The Crab Nebula (NGC 1952), the remains of the supernova of July 1054, an event observed and recorded at the Sung national observatory at K'ai-feng. In the intervening 900 years, the debris from the explosion has moved out about three lightyears; i.e., with a speed about 1 / 300 1 / 300 1//3001 / 3001/300 of that of light. In 1934 Walter Baade and Fritz Zwicky predicted that neutron stars should be produced in supernova explosions. Among the first half-dozen pulsars found in 1968 was one at the center of the Crab Nebula, pulsing 30 times per second, for which there is today no acceptable explanation other than a spinning neutron star. The Chinese historical record shown here lists unusual astronomical phenomena observed during the Northern Sung dynasty. It comes from the "Journal of Astronomy," part 9, chapter 56, of the Sung History (Sung Shih), first printed in the 1340 's. The photograph of that standard record used in this montage is copyright by, and may not be reproduced without permission of, the Trustees of the British Museum.

GRAVITATION

Charles W. MISNERKip S. THORNEJohn Archibald WHEELER

With a new foreword by
David I. KAISER
and a new preface by
Charles W. MISNER and Kip S. THORNE
Copyright © 2017 by Charles W. Misner, Kip S. Thorne, and the Estate of John Archibald Wheeler Foreword to the 2017 edition copyright © 2017 by Princeton University Press
Requests for permission to reproduce material from this work should be sent to Permissions, Princeton University Press
Published by Princeton University Press,
41 William Street, Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press,
6 Oxford Street, Woodstock, Oxfordshire OX20 1TR
press.princeton.edu
Cover illustration by Kenneth Gwin
Errata and additional information on sources may be found at the following webpage: http://press.princeton.edu/titles/11169.html
All Rights Reserved
ISBN 978-0-691-17779-3
Library of Congress Control Number: 2017950565
Original edition published in 1973 by W. H. Freeman and Company
First Princeton University Press edition, with a new foreword by David I. Kaiser and a new preface by Charles W. Misner and Kip S. Thorne, 2017
British Library Cataloging-in-Publication Data is available
Printed on acid-free paper. oo\infty
Printed in the United States of America
10987654321
We dedicate this book
To our fellow citizens
Who, for love of truth,
Take from their own wants
By taxes and gifts,
And now and then send forth
One of themselves
As dedicated servant,
To forward the search
Into the mysteries and marvelous simplicities
Of this strange and beautiful Universe,
Our home.

CONTENTS

LIST OF BOXES ..... xvii
LIST OF FIGURES ..... XX
FOREWORD TO THE 2017 EDITION ..... xxiii
PREFACE TO THE 2017 EDITION ..... xxxiii
PREFACE ..... xlvii
ACKNOWLEDGMENTS ..... li
Part I SPACETIME PHYSICS ..... 1
  1. Geometrodynamics in Brief ..... 3
  2. The Parable of the Apple ..... 3
  3. Spacetime With and Without Coordinates ..... 5
  4. Weightlessness ..... 13
  5. Local Lorentz Geometry, With and Without Coordinates ..... 19
    Time ..... 23
  6. Curvature ..... 29
    Effect of Matter on Geometry ..... 37
    Part II PHYSICS IN FLAT SPACETIME ..... 45
  7. Foundations of Special Relativity ..... 47
  8. Overview ..... 47
  9. Geometric Objects ..... 48
  10. Vectors ..... 49
  11. The Metric Tensor ..... 51
  12. Differential Forms ..... 53
  13. Gradients and Directional Derivatives ..... 59
  14. Coordinate Representation of Geometric Objects ..... 60 ..... 63
  15. The Centrifuge and the Photon
  16. Lorentz Transformations ..... 66
    0 0 0^(⇀)\stackrel{\rightharpoonup}{0}0 Collisions ..... 69
  17. The Electromagnetic Field ..... 71
  18. The Lorentz Force and the Electromagnetic Field Tensor ..... 71
  19. Tensors in All Generality ..... 74
  20. Three-Plus-One View Versus Geometric View ..... 78
  21. Maxwell's Equations ..... 79
    Working with Tensors ..... 81
  22. Electromagnetism and Differential Forms ..... 90
  23. Exterior Calculus ..... 90
  24. Electromagnetic 2-Form and Lorentz Force ..... 99
  25. Forms Illuminate Electromagnetism and Electromagnetism Illuminates Forms ..... 105
  26. Radiation Fields ..... 110
  27. Maxwell's Equations ..... 112
  28. Exterior Derivative and Closed Forms ..... 114
  29. Distant Action from Local Law ..... 120
  30. Stress-Energy Tensor and Conservation Laws ..... 130
  31. Track-1 Overview ..... 130
  32. Three-Dimensional Volumes and Definition of the Stress-Energy Tensor ..... 130
  33. Components of Stress-Energy Tensor ..... 137
  34. Stress-Energy Tensor for a Swarm of Particles ..... 138
  35. Stress-Energy Tensor for a Perfect Fluid ..... 139
  36. Electromagnetic Stress-Energy ..... 140
  37. Symmetry of the Stress-Energy Tensor ..... 141
  38. Conservation of 4-Momentum: Integral Formulation ..... 142
  39. Conservation of 4-Momentum: Differential Formulation ..... 146
  40. Sample Application of T = 0 T = 0 grad*T=0\boldsymbol{\nabla} \cdot \boldsymbol{T}=0T=0 ..... 152
  41. Angular Momentum ..... 156
  42. Accelerated Observers ..... 163
  43. Accelerated Observers Can Be Analyzed Using Special Relativity ..... 163
  44. Hyperbolic Motion ..... 166
  45. Constraints on Size of an Accelerated Frame ..... 168
  46. The Tetrad Carried by a Uniformly Accelerated Observer ..... 169
  47. The Tetrad Fermi-Walker Transported by an Observer with ArbitraryAcceleration 170
  48. The Local Coordinate System of an Accelerated Observer ..... 172
  49. Incompatibility of Gravity and Special Relativity ..... 177
  50. Attempts to Incorporate Gravity into Special Relativity ..... 177
  51. Gravitational Redshift Derived from Energy Conservation ..... 187
  52. Gravitational Redshift Implies Spacetime Is Curved ..... 187
  53. Gravitational Redshift as Evidence for the Principle of Equivalence ..... 189
  54. Local Flatness, Global Curvature ..... 190
    Part III THE MATHEMATICS OF CURVED SPACETIME ..... 193
  55. Differential Geometry: An Overview ..... 195
    An Overview of Part III 195
    1. Track 1 Versus Track 2: Difference in Outlook and Power ..... 197
Three Aspects of Geometry: Pictorial, Abstract, Component ..... 198
Tensor Algebra in Curved Spacetime ..... 201
5.Geodesics 207
6. Local Lorentz Frames: Mathematical Discussion 217Geodesic Deviation and the Riemann Curvature Tensor 218218
9. Differential Topology ..... 225
  1. Geometric Objects in Metric-Free, Geodesic-Free Spacetime ..... 225
  2. "Vector" and "Directional Derivative" Refined into Tangent Vector ..... 226
  3. Bases, Components, and Transformation Laws for Vectors ..... 230
  4. 1-Forms ..... 231
  5. Tensors ..... 233
  6. Commutators and Pictorial Techniques ..... 235
  7. Manifolds and Differential Topology ..... 240
  8. Affine Geometry: Geodesics, Parallel Transport and Covariant Derivative ..... 244
  9. Geodesics and the Equivalence Principle ..... 244
  10. Parallel Transport and Covariant Derivative: Pictorial Approach ..... 245
  11. Parallel Transport and Covariant Derivative: Abstract Approach ..... 247
  12. Parallel Transport and Covariant Derivative: Component Approach ..... 258
  13. Geodesic Equation ..... 262
  14. Geodesic Deviation and Spacetime Curvature ..... 265
  15. Curvature, At Last! ..... 265
  16. The Relative Acceleration of Neighboring Geodesics ..... 265
  17. Tidal Gravitational Forces and Riemann Curvature Tensor ..... 270
  18. Parallel Transport Around a Closed Curve ..... 277
  19. Flatness is Equivalent to Zero Riemann Curvature ..... 283
  20. Riemann Normal Coordinates ..... 285
  21. Newtonian Gravity in the Language of Curved Spacetime ..... 289
  22. Newtonian Gravity in Brief ..... 289
  23. Stratification of Newtonian Spacetime ..... 291
  24. Galilean Coordinate Systems ..... 292
  25. Geometric, Coordinate-Free Formulation of Newtonian Gravity ..... 298
  26. The Geometric View of Physics: A Critique ..... 302
  27. Riemannian Geometry: Metric as Foundation of All ..... 304
  28. New Features Imposed on Geometry by Local Validity of Special Relativity 304
  29. Metric ..... 305
  30. Concord Between Geodesics of Curved Spacetime Geometry and Straight Lines of Local Lorentz Geometry ..... 312
  31. Geodesics as World Lines of Extremal Proper Time ..... 315
  32. Metric-Induced Properties of Riemann ..... 324
  33. The Proper Reference Frame of an Accelerated Observer ..... 327
  34. Calculation of Curvature ..... 333
  35. Curvature as a Tool for Understanding Physics ..... 333
  36. Forming the Einstein Tensor ..... 343
  37. More Efficient Computation ..... 344
  38. The Geodesic Lagrangian Method ..... 344
  39. Curvature 2-Forms ..... 348
  40. Computation of Curvature Using Exterior Differential Forms ..... 354
  41. Bianchi Identities and the Boundary of a Boundary ..... 364
  42. Bianchi Identities in Brief ..... 364
  43. Bianchi Identity d % = 0 d % = 0 d%=0\boldsymbol{d} \%=0d%=0 as a Manifestation of "Boundary ofBoundary = 0 = 0 =0=0=0 "372
  44. Moment of Rotation: Key to Contracted Bianchi Identity ..... 373
  45. Calculation of the Moment of Rotation ..... 375
  46. Conservation of Moment of Rotation Seen from "Boundary of a Boundary is Zero" 377
  47. Conservation of Moment of Rotation Expressed in Differential Form ..... 378
  48. From Conservation of Moment of Rotation to Einstein's Geometrodynamics: A Preview ..... 379
    Part IV EINSTEIN'S GEOMETRIC THEORY OF GRAVITY ..... 383
  49. Equivalence Principle and Measurement of the "Gravitational Field" ..... 385
  50. Overview ..... 385
    The Laws of Physics in Curved Spacetime ..... 385
    Factor-Ordering Problems in the Equivalence Principle ..... 388
  51. The Rods and Clocks Used to Measure Space and Time Intervals ..... 393
    The Measurement of the Gravitational Field ..... 399
  52. How Mass-Energy Generates Curvature ..... 404
  53. Automatic Conservation of thof the Field Equation 404
  54. Automatic Conservation of the Source: A Dynamic Necessity ..... 408 ..... 409
    Cosmological Constant
    Cosmological Constant
  55. The Newtonian Limit ..... 412
    Axiomatize Einstein's Theory? ..... 416
  56. "No Prior Geometry": A Feature Distinguishing Einstein's Theory from Other Theories of Gravity ..... 429
  57. A Taste of the History of Einstein's Equation ..... 431
  58. Weak Gravitational Fields ..... 435
  59. The Linearized Theory of Gravity ..... 435
    Gravitational Waves ..... 442
  60. Effect of Gravity on Matter ..... 442
    Nearly Newtonian Gravitational Fields ..... 445
  61. Mass and Angular Momentum of a Gravitating System ..... 448
  62. External Field of a Weakly Gravitating Source ..... 448
  63. Measurement of the Mass and Angular Momentum ..... 450
  64. Mass and Angular Momentum of Fully Relativistic Sources ..... 451
  65. Mass and Angular Momentum of a Closed Universe ..... 457
  66. Conservation Laws for 4-Momentum and Angular Momentum ..... 460
  67. Overview ..... 460
  68. Gaussian Flux Integrals for 4-Momentum and Angular Momentum ..... 461
  69. Volume Integrals for 4-Momentum and Angular Momentum ..... 464
  70. Why the Energy of the Gravitational Field Cannot be Localized ..... 466
  71. Conservation Laws for Total 4-Momentum and Angular Momentum ..... 468
  72. Equation of Motion Derived from the Field Equation ..... 471
  73. Variational Principle and Initial-Value Data ..... 484
  74. Dynamics Requires Initial-Value Data ..... 484
  75. The Hilbert Action Principle and the Palatini Method of Variation ..... 491
  76. Matter Lagrangian and Stress-Energy Tensor ..... 504
  77. Splitting Spacetime into Space and Time ..... 505
  78. Intrinsic and Extrinsic Curvature ..... 509
  79. The Hilbert Action Principle and the Arnowitt-Deser-Misner Modification Thereof in the Space-plus-Time Split ..... 519
  80. The Arnowitt-Deser-Misner Formulation of the Dynamics of Geometry ..... 520
  81. Integrating Forward in Time ..... 526
  82. The Initial-Value Problem in the Thin-Sandwich Formulation ..... 528
  83. The Time-Symmetric and Time-Antisymmetric Initial-Value Problem ..... 535
  84. York's "Handles" to Specify a 4-Geometry ..... 539
  85. Mach's Principle and the Origin of Inertia ..... 543
  86. Junction Conditions ..... 551

22. Thermodynamics, Hydrodynamics, Electrodynamics, Geometric Optics, and Kinetic Theory 557

  1. The Why of this Chapter 557
  2. Thermodynamics in Curved Spacetime 557
  3. Hydrodynamics in Curved Spacetime 562
  4. Electrodynamics in Curved Spacetime 568
  5. Geometric Optics in Curved Spacetime 570
  6. Kinetic Theory in Curved Spacetime 583

Part V RELATIVISTIC STARS 591

  1. Spherical Stars 593
  2. Prolog 593
  3. Coordinates and Metric for a Static, Spherical System 594
  4. Physical Interpretation of Schwarzschild coordinates 595
  5. Description of the Matter Inside a Star 597
  6. Equations of Structure 600
  7. External Gravitational Field 607
  8. How to Construct a Stellar Model 608
  9. The Spacetime Geometry for a Static Star 612
  10. Pulsars and Neutron Stars; Quasars and Supermassive Stars 618
  11. Overview 618
  12. The Endpoint of Stellar Evolution 621
  13. Pulsars 627
  14. Supermassive Stars and Stellar Instabilities 630
  15. Quasars and Explosions In Galactic Nuclei 634
  16. Relativistic Star Clusters 634
  17. The "Pit in the Potential" as the Central New Feature of Motion in
    Schwarzschild Geometry 636
  18. From Kepler's Laws to the Effective Potential for Motion in Schwarzschild Geometry 636
  19. Symmetries and Conservation Laws 650
  20. Conserved Quantities for Motion in Schwarzschild Geometry 655
  21. Gravitational Redshift 659
  22. Orbits of Particles 659
  23. Orbit of a Photon, Neutrino, or Graviton in Schwarzschild Geometry 672
  24. Spherical Star Clusters 679
  25. Stellar Pulsations 688
  26. Motivation 688
  27. Setting Up the Problem 689
  28. Eulerian versus Lagrangian Perturbations 690
  29. Initial-Value Equations 691
  30. Dynamic Equation and Boundary Conditions 693
  31. Summary of Results 694

Part VI THE UNIVERSE
701

27. Idealized Cosmologies
703

1.I The Homogeneity and Isotropy of the Universe 703
2. Stress-Energy Content of the Universe-the Fluid Idealization 711
3. Geometric Implications of Homogeneity and Isotropy 713
4. Comoving, Synchronous Coordinate Systems for the Universe ..... 715
5. The Expansion Factor ..... 718
6. Possible 3-Geometries for a Hypersurface of Homogeneity ..... 720
7. Equations of Motion for the Fluid ..... 726
8. The Einstein Field Equations ..... 728
9. Time Parameters and the Hubble Constant ..... 730
10. The Elementary Friedmann Cosmology of a Closed Universe ..... 733
11. Homogeneous Isotropic Model Universes that Violate Einstein's Conception of Cosmology ..... 742
28. Evolution of the Universe into Its Present State ..... 763
  1. The "Standard Model" of the Universe ..... 763
  2. Standard Model Modified for Primordial Chaos ..... 769
  3. What "Preceded" the Initial Singularity? ..... 769
  4. Other Cosmological Theories ..... 770
  5. Present State and Future Evolution of the Universe ..... 771
  6. Parameters that Determine the Fate of the Universe ..... 771
  7. Cosmological Redshift ..... 772
  8. The Distance-Redshift Relation: Measurement of the Hubble Constant ..... 780
  9. The Magnitude-Redshift Relation: Measurement of the Deceleration Parameter ..... 782
  10. Search for "Lens Effect" of the Universe ..... 795
  11. Density of the Universe Today ..... 796
  12. Summary of Present Knowledge About Cosmological Parameters ..... 797
  13. Anisotropic and Inhomogeneous Cosmologies ..... 800
  14. Why Is the Universe So Homogeneous and Isotropic? ..... 800
  15. The Kasner Model for an Anisotropic Universe ..... 801
  16. Adiabatic Cooling of Anisotropy ..... 802
  17. Viscous Dissipation of Anisotropy ..... 802
  18. Particle Creation in an Anisotropic Universe ..... 803
  19. Inhomogeneous Cosmologies ..... 804
  20. The Mixmaster Universe ..... 805
  21. Horizons and the Isotropy of the Microwave Background ..... 815
    Part VII GRAVITATIONAL COLLAPSE AND BLACK HOLES ..... 817
  22. Schwarzschild Geometry ..... 819
  23. Inevitability of Collapse for Massive Stars ..... 819
  24. The Nonsingularity of the Gravitational Radius ..... 820
  25. Behavior of Schwarzschild Coordinates at r = 2 M r = 2 M r=2Mr=2 Mr=2M ..... 823
  26. Several Well-Behaved Coordinate Systems ..... 826
  27. Relationship Between Kruskal-Szekeres Coordinates and Schwarzschild Coordinates ..... 833
  28. Dynamics of the Schwarzschild Geometry ..... 836
  29. Gravitational Collapse ..... 842
  30. Relevance of Schwarzschild Geometry ..... 842
  31. Birkhoff's Theorem ..... 843
  32. Exterior Geometry of a Collapsing Star ..... 846
  33. Collapse of a Star with Uniform Density and Zero Pressure ..... 851
  34. Spherically Symmetric Collapse with Internal Pressure Forces ..... 857
  35. The Fate of a Man Who Falls into the Singularity at r = 0 r = 0 r=0r=0r=0 ..... 860
  36. Realistic Gravitational Collapse-An Overview ..... 862

33. Black Holes 872

  1. Why "Black Hole"? 872
  2. The Gravitational and Electromagnetic Fields of a Black Hole 875
  3. Mass, Angular Momentum, Charge, and Magnetic Moment 891
  4. Symmetries and Frame Dragging 892
  5. Equations of Motion for Test Particles 897
  6. Principal Null Congruences 901
  7. Storage and Removal of Energy from Black Holes 904
  8. Reversible and Irreversible Transformations 907
  9. Global Techniques, Horizons, and Singularity Theorems 916
  10. Global Techniques Versus Local Techniques 916
  11. "Infinity" in Asymptotically Flat Spacetime 917
  12. Causality and Horizons 922
  13. Global Structure of Horizons 924
  14. Proof of Second Law of Black-Hole Dynamics 931
  15. Singularity Theorems and the "Issue of the Final State" 934

Part VIII GRAVITATIONAL WAVES 941

35. Propagation of Gravitational Waves 943

  1. Viewpoints 943
  2. Review of "Linearized Theory" in Vacuum 944
  3. Plane-Wave Solutions in Linearized Theory 945
  4. The Transverse Traceless (TT) Gauge 946
  5. Geodesic Deviation in a Linearized Gravitational Wave 950
6 Polarization of a Plane Wave 952
7. The Stress-Energy Carried by a Gravitational Wave 955
8. Gravitational Waves in the Full Theory of General Relativity 956
9. An Exact Plane-Wave Solution 957
10. Physical Properties of the Exact Plane Wave 960
11. Comparison of an Exact Electromagnetic Plane Wave with the Gravitational Plane Wave 961
12. A New Viewpoint on the Exact Plane Wave 962
13. The Shortwave Approximation 964
14. Effect of Background Curvature on Wave Propagation 967
15. Stress-Energy Tensor for Gravitational Waves 969
36. Generation of Gravitational Waves 974
  1. The Quadrupole Nature of Gravitational Waves 974
  2. Power Radiated in Terms of Internal Power Flow 978
  3. Laboratory Generators of Gravitational Waves 979
  4. Astrophysical Sources of Gravitational Waves: General Discussion 980
Gravitational Collapse, Black Holes, Supernovae, and Pulsars as Sources
6. Binary Stars as Sources 986
7. Formulas for Radiation from Nearly Newtonian Slow-Motion Sources 989
8. Radiation Reaction in Slow-Motion Sources 993
9. Foundations for Derivation of Radiation Formulas 995
10. Evaluation of the Radiation Field in the Slow-Motion Approximation 996
11. Derivation of the Radiation-Reaction Potential 1001
37. Detection of Gravitational Waves 1004
  1. Coordinate Systems and Impinging Waves 1004
  2. Accelerations in Mechanical Detectors 1006
  3. Types of Mechanical Detectors 1012
  4. Vibrating, Mechanical Detectors: Introductory Remarks ..... 1019
  5. Idealized Wave-Dominated Detector, Excited by Steady Flux of MonochromaticWaves1022
  6. Idealized, Wave-Dominated Detector, Excited by Arbitrary Flux of Radiation ..... 1026
  7. General Wave-Dominated Detector, Excited by Arbitrary Flux ofRadiation 1028
  8. Noisy Detectors ..... 1036
  9. Nonmechanical Detectors ..... 1040
  10. Looking Toward the Future ..... 1040
    Part IX. EXPERIMENTAL TESTS OF GENERAL RELATIVITY ..... 1045
  11. Testing the Foundations of Relativity ..... 1047
  12. Testing is Easier in the Solar System than in Remote Space ..... 1047
    Theoretical Frameworks for Analyzing Tests of General Relativity ..... 1048
  13. Tests of the Principle of the Uniqueness of Free Fall: Eötvös-Dicke Experiment ..... 1050
  14. Tests for the Existence of a Metric Governing Length and Time Measurements ..... 1054
  15. Tests of Geodesic Motion: Gravitational Redshift Experiments ..... 1055
  16. Tests of the Equivalence Principle ..... 1060
  17. Tests for the Existence of Unknown Long-Range Fields ..... 1063
  18. Other Theories of Gravity and the Post-Newtonian Approximation ..... 1066
  19. Other Theories ..... 1066
  20. Metric Theories of Gravity ..... 1067
  21. Post-Newtonian Limit and PPN Formalism ..... 1068
  22. PPN Coordinate System ..... 1073
  23. Description of the Matter in the Solar System ..... 1074
  24. Nature of the Post-Newtonian Expansion ..... 1075
  25. Newtonian Approximation ..... 1077
  26. PPN Metric Coefficients ..... 1080
  27. Velocity of PPN Coordinates Relative to "Universal Rest Frame" ..... 1083
  28. PPN Stress-Energy Tensor ..... 1086
  29. PPN Equations of Motion ..... 1087
  30. Relation of PPN Coordinates to Surrounding Universe ..... 1091
  31. Summary of PPN Formalism ..... 109
  32. Solar-System Experiments ..... 1096
  33. Many Experiments Open to Distinguish General Relativity from Proposed Metric Theories of Gravity 1096
  34. The Use of Light Rays and Radio Waves to Test Gravity ..... 1099
  35. "Light" Deflection ..... 1101
  36. Time-Delay in Radar Propagation ..... 1103
  37. Perihelion Shift and Periodic Perturbations in Geodesic Orbits ..... 1110
  38. Three-Body Effects in the Lunar Orbit ..... 1116
  39. The Dragging of Inertial Frames ..... 1117
  40. Is the Gravitational Constant Constant? ..... 1121
  41. Do Planets and the Sun Move on Geodesics? ..... 1126
  42. Summary of Experimental Tests of General Relativity ..... 1131
    Part X. FRONTIERS ..... 1133
  43. Spinors ..... 1135
  44. Reflections, Rotations, and the Combination of Rotations ..... 1135
  45. Infinitesimal Rotations ..... 1140
  46. Lorentz Transformation via Spinor Algebra ..... 1142
  47. Thomas Precession via Spinor Algebra ..... 1145
  48. Spinors ..... 1148
  49. Correspondence Between Vectors and Spinors ..... 1150
  50. Spinor Algebra ..... 1151
  51. Spin Space and Its Basis Spinors ..... 1156
  52. Spinor Viewed as Flagpole Plus Flag Plus Orientation-Entanglement Relation ..... 1157
  53. Appearance of the Night Sky: An Application of Spinors ..... 1160
  54. Spinors as a Powerful Tool in Gravitation Theory ..... 1164
  55. Regge Calculus ..... 1166
  56. Why the Regge Calculus? ..... 1166
  57. Regge Calculus in Brief ..... 1166
  58. Simplexes and Deficit Angles ..... 1167
  59. Skeleton Form of Field Equations ..... 1169
  60. The Choice of Lattice Structure ..... 1173
  61. The Choice of Edge Lengths ..... 1177
  62. Past Applications of Regge Calculus ..... 1178
  63. The Future of Regge Calculus ..... 1179
  64. Superspace: Arena for the Dynamics of Geometry ..... 1180
  65. Space, Superspace, and Spacetime Distinguished ..... 1180
  66. The Dynamics of Geometry Described in the Language of the Superspace of the (3) ^('){ }^{\prime} 's ..... 1184
  67. The Einstein-Hamilton-Jacobi Equation ..... 1185
  68. Fluctuations in Geometry ..... 1190
  69. Beyond the End of Time ..... 1196
  70. Gravitational Collapse as the Greatest Crisis in Physics of All Time ..... 1196
  71. Assessment of the Theory that Predicts Collapse ..... 1198
  72. Vacuum Fluctuations: Their Prevalence and Final Dominance ..... 1202
  73. Not Geometry, but Pregeometry, as the Magic Building Materia ..... 1203
  74. Pregeometry as the Calculus of Propositions ..... 1208
  75. The Black Box: The Reprocessing of the Universe ..... 1209
    Bibliography and Index of Names ..... 1221
    Subject Index ..... 1255

BOXES

1.1. Mathematical notation for events, coordinates, and vectors. 9
1.2. Acceleration independent of composition. 16
1.3. Local Lorentz and local Euclidean geometry. 20
1.4. Time today. 28
1.5. Test for flatness. 30
1.6. Curvature of what? 32
1.7. Lorentz force equation and geodesic deviation equation compared. 35
1.8. Geometrized units 36
1.9. Galileo Galilei. 38
1.10. Isaac Newton. 40
1.11. Albert Einstein. 42
2.1. Farewell to "ict." 51
2.2. Worked exercises using the metric. 54
2.3. Differentials 63
2.4. Lorentz transformations 67
3.1. Lorentz force law defines fields, predicts motions. 72
3.2. Metric in different languages. 77
3.3. Techniques of index gymnastics. 85
4.1. Differential forms and exterior calculus in brief. 91
4.2. From honeycomb to abstract 2-form. 102
4.3. Duality of 2 -forms. 108
4.4. Progression of forms and exterior derivatives. 115
4.5. Metric structure versus Hamiltonian or symplectic structure. 126
4.6. Birth of Stokes' Theorem. 127
5.1. Stress-energy summarized. 131
5.2. Three-dimensional volumes. 135
5.3. Volume integrals, surface integrals, and Gauss's theorem in component notation. 147
5.4. Integrals and Gauss's theorem in the language of forms. 150
5.5. Newtonian hydrodynamics reviewed. 153
5.6. Angular momentum. 157
6.1. General relativity built on special relativity. 164
6.2. Accelerated observers in brief. 164
7.1. An attempt to describe gravity as a symmetric tensor field in flat spacetime. 181
8.1. Books on differential geometry. 196
8.2. Elie Cartan. 198
8.3. Pictorial, abstract, and component treatments of differential geometry. 199
8.4. Local tensor algebra in an arbitrary basis. 202
8.5. George Friedrich Bernhard Riemann. 220
8.6. Fundamental equations for covariant derivative and curvature. 223
9.1. Tangent vectors and tangent space. 227
9.2. Commutator as closer of quadrilaterals. 236
10.1. Geodesics. 246
10.2. Parallel transport and covariant differentiation in terms of Schild's ladder. 248
10.3. Covariant derivative: the machine and its components. 254
11.1. Geodesic deviation and curvature in brief. 266
11.2. Geodesic deviation represented as an arrow 268
11.3. Arrow correlated with second derivative. 270
11.4. Newtonian and geometric analyses of relative acceleration. 272
11.5. Definition of Riemann curvature tensor. 273
11.6. Geodesic deviation and parallel transport around a closed curve as two aspects of same construction. 279
11.7. The law for parallel transport around a closed curve. 281
12.1. Geodesic deviation in Newtonian spacetime. 293
12.2. Spacetimes of Newton, Minkowski, and Einstein. 296
12.3. Treatments of gravity of Newton à la Cartan and of Einstein. 297

12.4. Geometric versus standard formulation of Newtonian gravity. 300

13.1. Metric distilled from distances. 306
13.2. "Geodesic" versus "extremal world line." 322
13.3. "Dynamic" variational principle for geodesics. 322
14.1. Perspectives on curvature. 335
14.2. Straightforward curvature computation. 340
14.3. Analytical calculations on a computer. 342
14.4. Geodesic Lagrangian method shortens some curvature computations. 346
14.5. Curvature computed using exterior differential
forms (metric for Friedmann cosmology). 355
15.1. The boundary of a boundary is zero. 365
15.2. Mathematical representations for the moment of rotation and the source of gravitation. 379
15.3. Other identities satisfied by the curvature. 381
16.1. Factor ordering and coupling to curvature in applications of the equivalence principle. 390
16.2. Pendulum clock analyzed. 394
16.3. Response of clocks to acceleration. 396
16.4. Ideal rods and clocks built from geodesic world lines. 397
16.5. Gravity gradiometer for measuring Riemann curvature. 401
17.1. Correspondence principles. 412
17.2. Six routes to Einstein's geometrodynamic law. 417
17.3. An experiment on prior geometry. 430
18.1. Derivations of general relativity from geometric viewpoint and from theory of field of spin two. 437
18.2. Gauge transformations and coordinate transfor-
19.1. Mass-energy, 4-momentum, and angular momentum of an isolated system. 454
19.2. Metric correction term near selected heavenly bodies. 459
20.1. Proper Lorentz transformation and duality rotation. 482
20.2. Transformation of generic electromagnetic field
tensor in local inertial frame. 483
21.1. Hamiltonian as dispersion relation. 493
21.2. Counting the degrees of freedom of the electromagnetic field. 530
22.1. Alternative thermodynamic potentials. 561
22.2. Thermodynamics and hydrodynamics of a perfect fluid in curved spacetime. 564
22.3. Geometry of an electromagnetic wavetrain.
574
22.4. Geometric optics in curved spacetime. 578
22.5. Volume in phase space. 585
22.6. Conservation of volume in phase space. 586
23.1. Mass-energy inside radius r r rrr. 603
23.2. Model star of uniform density. 609
23.3. Rigorous derivation of the spherically symmetric line element. 616
24.1. Stellar configurations where relativistic effects are important. 619
24.2. Oscillation of a Newtonian star. 630
25.1. Mass from mean angular frequency and semimajor axis. 638
25.2. Motion in Schwarzschild geometry as point of departure for major applications of Einstein's theory. 640
25.3. Hamilton-Jacobi description of motion: natural because ratified by quantum principle. 641
25.4. Motion in Schwarzschild geometry analyzed by Hamilton-Jacobi method. 644
25.5. Killing vectors and isometries. 652
25.6. Motion of a particle in Schwarzschild geometry. 660
25.7. Motion of a photon in Schwarzschild geometry. 674
25.8. Equations of structure for a spherical star cluster. 683
25.9. Isothermal star clusters. 685
26.1. Eigenvalue problem and variational principle for normal-mode pulsations. 695
26.2. Critical adiabatic index for nearly Newtonian stars. 697
27.1. Cosmology in brief. 704
27.2. The 3 -geometry of hypersurfaces of homogeneity. 723
27.3. Friedmann cosmology for matter-dominated and radiation-dominated model universes. 734
27.4. A typical cosmological model that agrees with astronomical observations. 738
27.5. Effect of choice of Λ Λ Lambda\LambdaΛ and choice of closed or open on the predicted course of cosmology. 746
27.6. Alexander Alexandrovitch Friedmann. 751
27.7. A short history of cosmology. 752
28.1. Evolution of the quasar population. 767
29.1. Observational parameters compared to relativity pararieters. 773
29.2. Redshift of the primordial radiation. 779
29.3. Use of redshift to characterize distance and time. 779
29.4. Measurement of Hubble constant and deceleration parameter. 785
29.5. Edwin Powell Hubble. 792
30.1. The mixmaster universe. 806
31.1. The Schwarzschild singularity: historical remarks. 822
31.2. Motivation for Kruskal-Szerekes coordinates. 828
32.1. Collapsing star with Friedmann interior and Schwarschild exterior. 854
32.2. Collapse with nonspherical perturbations. 864
32.3. Collapse in one and two dimensions. 867
33.1. A black hole has no hair. 876
33.2. Kerr-Newman geometry and electromagnetic field. 878
33.3. Astrophysics of black holes. 883
33.4. The laws of black-hole dynamics. 887
33.5. Orbits in "equatorial plane" of Kerr-Newman black hole. 911
34.1. Horizons are generated by nonterminating null geodesics. 926
34.2. Roger Penrose. 936
34.3. Stephen W. Hawking. 938
35.1. Transverse-traceless part of a wave. 948
36.1. Gravitational waves from pulsating neutron stars. 984
36.2. Analysis of burst of radiation from impulse event. 987
36.3. Radiation from several binary star systems. 990
37.1. Derivation of equations of motion of detector. 1007
37.2. Lines of force for gravitational-wave accelerations. 1011
37.3. Use of cross-section for wave-dominated detector. 1020
37.4. Vibrating, resonant detector of arbitrary shape. 1031
37.5. Detectability of hammer-blow waves from astrophysical sources. 1041.
37.6. Nonmechanical detector. 1043
38.1. Technology of the 1970's confronted with relativistic phenomena. 1048
38.2. Baron Lorand von Eötvös. 1051
38.3. Robert Henry Dicke. 1053
39.1. The theories of Dicke-Brans-Jordan and of Ni. 1070
39.2. Heuristic description of the ten post-Newtonian parameters. 1072
39.3. Post-Newtonian expansion of the metric coefficients. 1077
39.4. Summary of the PPN formalism. 1092
39.5. PPN parameters used in the literature: a translator's guide. 1093
40.1. Experimental results on deflection of light and radio waves. 1104
40.2. Experimental results on radar time-delay. 1109
40.3. Experimental results on perihelion precession. 1112
40.4. Catalog of experiments. 1129
41.1. Spinor representation of simple tensors. 1154
42.1. The hinges where "angle of rattle" is concentrated in two, three, and four dimensions. 1169
42.2. Flow diagrams for Regge calculus. 1171
42.3. Synthesis of higher-dimensional skeleton geometries out of lower-dimensional ones. 1176
43.1. Geometrodynamics compared with particle dynamics. 1181
44.1. Collapse of universe compared and contrasted with collapse of atom. 1197
44.2. Three levels of gravitational collapse. 1201
44.3. Relation of spin 1 2 1 2 (1)/(2)\frac{1}{2}12 to geometrodynamics. 1204
44.4. Bucket-of-dust concept of pregeometry. 1205
44.5. Pregeometry as the calculus of propositions. 1211

FIGURES

1.1. Spacetime compared with the surface of an apple. 4
1.2. World-line crossings mark events. 6
1.3. Two systems of coordinates for same events. 7
1.4. Mere coordinate singularities. 11
1.5. Singularities in the coordinates on a 2 -sphere. 12
1.6. The Roll-Krotkov-Dicke experiment. 14
1.7. Testing for a local inertial frame. 18
1.8. Path of totality of an ancient eclipse. 25
1.9. Good clock versus bad clock. 27
1.10. "Acceleration of the separation" of nearby geodesics. 31
1.11. Separation of geodesics in a 3-manifold. 31
1.12. Satellite period and Earth density. 39
2.1. From bilocal vector to tangent vector. 49
2.2. Different curves, same tangent vector. 50
2.3. Velocity 4 -vector resolved into components. 52
2.4. A 1 -form pierced by a vector. 55
2.5. Gradient as 1 -form. 56
2.6. Addition of 1 -forms. 57
2.7. Vectors and their corresponding 1 -forms. 58
2.8. Lorentz basis. 60
2.9. The centrifuge and the photon. 63
4.1. Faraday 2 -form. 100
4.2. Faraday form creates a 1 -form out of 4 -velocity. 104
4.3. Spacelike slices through Faraday. 106
4.4. Faraday and its dual, Maxwell. 107
4.5. Maxwell 2 -form for charge at rest. 109
4.6. Mechanism of radiation. 111
4.7. Simple types of 1 -form. 123
5.1. River of 4 -momentum sensed by different 3 volumes. 133
5.2. Aluminum ring lifted by Faraday stresses. 141
5.3. Integral conservation laws for energy-momentum. 143
6.1. Hyperbolic motion. 167
6.2. World line of accelerated observer. 169
6.3. Hyperplanes orthogonal to curved world
line. 172
6.4. Local coordinates for observer in hyperbolic motion. 173
7.1. Congruence of world lines of successive light pulses. 188
8.1. Basis vectors for Kepler orbit. 200
8.2. Covariant derivation. 209
8.3. Connection coefficients as aviator's turning coefficients. 212
8.4. Selector parameter and affine parameter for a family of geodesics. 219
9.1. Basis vectors induced by a coordinate system. 231
9.2. Basis vectors and dual basis 1 -forms. 232
9.3. Three representations of S 2 .241 S 2 .241 S^(2).241S^{2} .241S2.241
10.1. Straight-on parallel transport. 245
10.2. Nearby tangent spaces linked by parallel transport. 252
11.1. One-parameter family of geodesics. 267
11.2. Parallel transport around a closed curve. 278
12.1. Coordinates carried by an Earth satellite. 298
13.1. Distances determine geometry. 309
13.2. Two events connected by more than one geodesic. 318
13.3. Coordinates in the truncated space of all histories. 320
13.4. Proper reference frame of an accelerated observer. 328
15.1. The rotations associated with all six faces add to zero. 372
18.1. Primitive detector for gravitational waves. 445
20.1. "World tube." 473
20.2. "Buffer zone." 477
21.1. Momentum and energy as rate of change of "dy
namic phase." 487
21.2. Building a thin-sandwich 4 -geometry. 506
21.3. Extrinsic curvature. 511
21.4. Spacelike slices through Schwarzschild geome
try. 528
21.5. Einstein thanks Mach. 544
21.6. Gaussian normal coordinates. 552
22.1. Geometric optics for a bundle of rays. 581
22.2. Number density of photons and specific intensity. 589
23.1. Geometry within and around a star. 614
24.1.| First publications on black holes and neutron stars. 622
24.2. Harrison-Wheeler equation of state for cold catalyzed matter and Harrison-Wakano-Wheeler stellar models. 625
24.3. Collapse, pursuit, and plunge scenario. 629
25.1. Jupiter's satellites followed from night to night. 637
25.2. Effective potential for motion in Schwarzschild geometry. 639
25.3. Cycloid relation between r r rrr and t t ttt for straight-in fall. 664
25.4. Effective potential as a function of the tortoise coordinate. 666
25.5. Fall toward a black hole as described by a comoving observer versus a far-away observer. 667
25.6. I Photon orbits in Schwarzschild geometry. 677
25.7. Deflection of a photon as a function of impact parameter. 678
27.1. Comoving, synchronous coordinate system for the universe. 716
27.2. Expanding balloon models an expanding universe. 719
27.3. Schwarzschild zones fitted together to make a closed universe. 739
27.4. Friedmann cosmology in terms of arc parameter time and hyperpolar angle. 741
27.5. Effective potential for Friedmann dynamics. 748
28.1. Temperature and density versus time for the standard big-bang model. 764
29.1. Redshift as an effect of standing waves. 776
29.2. Angle-effective distance versus redshift. 796
31.1. Radial geodesics charted in Schwarzschild coordinates. 825
31.2. Novikov coordinates for Schwarzschild geometry. 827
31.3. Transformation from Schwarzschild to KruskalSzekeres coordinates. 834
31.4. Varieties of radial geodesic presented in Schwarzschild and Kruskal-Szekeres coordinates. 835
31.5. Embedding diagram for Schwarzschild geometry at a moment of time symmetry. 837
31.6. Dynamics of the Schwarzschild throat. 839
32.1. Free-fall collapse of a star. 848
33.1. Surface of last influence for collapsing star. 873
33.2. Black hole as garbage dump and energy source. 908
33.3. Energy of particle near a Kerr black hole. 910
34.1. Future null infinity and the energy radiated in a supernova explosion. 918
34.2. Minkowski spacetime depicted in coordinates that are finite at infinity. 919
34.3. Schwarzschild spacetime . 920
34.4 Reissner-Nordstrøm spacetime depicted in coordinates that are finite at infinity. 921
34.5. Spacetime diagrams for selected causal relationships. 922
34.6. Black holes in an asymptotically flat spacetime. 924
34.7. The horizon produced by spherical collapse of a star. 925
34.8. Spacetime diagram used to prove the second law of black-hole dynamics. 932
35.1. Plane electromagnetic waves. 952
35.2. Plane gravitational waves. 953
35.3. Exact plane-wave solution. 959
36.1. Why gravitational radiation is ordinarily weak. 976
36.2. Spectrum given off in head-on plunge into a Schwarzschild black hole. 983
36.3. Slow-motion source. 997
37.1. Reference frame for vibrating bar detector. 1005
37.2. Types of detectors. 1013
37.3. Separation between geodesics responds to a gravitational wave. 1014
37.4. Vibrator responding to linearly polarized radiation. 1022
37.5. Hammer blow of a gravitational wave on a noisy detector. 1037
38.1. The Pound-Rebka-Snider measurement of gravitational redshift on the Earth. 1057
38.2. Brault's determination of the redshift of the D 1 D 1 D_(1)D_{1}D1 line of sodium from the sun. 1059
38.3. The Turner-Hill search for a dependence of proper clock rate on velocity relative to distant matter. 1065
40.1. Bending of trajectory near the sun. 1100
40.2. Coordinates used in calculating the deflection of light. 1101

40.3. Coordinates for calculating the relativistic timedelay. 1106

40.4. Laser measurement of Earth-moon separation. 1130

41.1. Combination of rotations of 90 90 90^(@)90^{\circ}90 about axes that

diverge by 90 90 90^(@)90^{\circ}90. 1136
41.2. Rotation depicted as two reflections. 1137
41.3. Composition of two rotations seen in terms of reflections. 1138
41.4. Law of composition of rotations epitomized in a spherical triangle. 1139
41.5. "Orientation-entanglement relation" between a cube and its surroundings. 1148
41.6. A 720 720 720^(@)720^{\circ}720 rotation is equivalent to no rotation. 1149
41.7. Spinor as flagpole plus flag. 1157
41.8. Direction in space represented on the complex plane. 1161
42.1. A 2-geometry approximated by a polyhedron. 1168
42.2. Cycle of building blocks associated with a single hinge. 1170
43.1. Superspace in the simplicial approximation. 1182
43.2. Space, spacetime, and superspace. 1183
43.3. Electron motion affected by field fluctuations. 1190
44.1. Wormhole picture of electric charge. 1200
44.2. Gravitation as the metric elasticity of space. 1207
44.3. What pregeometry is not. 1210
44.4. Black-box model for reprocessing of universe. 1213
44.5. A mind full of geometrodynamics. 1219

FOREWORD TO THE 2017 PRINTING OF GRAVITATION
DAVID I. KAISER

A remarkable publishing event occurred in September 1973: the release of a 1,279-page book, weighing more than six pounds, with the simple title, Gravitation. 1 1 ^(1){ }^{1}1 Wags were quick to remark that the book was not just about gravitation, but a significant source of it. The book acquired several nicknames, including "the phone book" (another reference to its girth) and "the big black book" (for its sleek, modern cover). Most common became "MTW," named for the authors' initials: Charles Misner, Kip Thorne, and John Wheeler. 2 2 ^(2){ }^{2}2
Gravitation focuses on the general theory of relativity, Albert Einstein's remarkable theory of gravity. Einstein completed a version of this theory, in a form we would recognize today, just over a century ago, presenting the finishing touches in a flurry of brief communications to the Prussian Academy of Sciences in November 1915. His major insight was that space and time were actors in the story of nature, not merely a fixed stage on which all other activity played out. Space and time, on Einstein's account, were dynamical-they could bend and distend in response to the distribution of matter and
energy. That warping, in turn, would affect objects' motion, diverting them from the straight and narrow path. 3 3 ^(3){ }^{3}3
One year after the armistice that ended the First World War, a British team, led by Arthur Eddington, announced that they had confirmed one of Einstein's key predictions: that gravity could bend the path of starlight. The dramatic announcement propelled Einstein and his general theory to instant stardom. Yet interest in the theory waned over the 1930s. Einstein himself noted plaintively, in a preface for a colleague's textbook in 1942, "I believe that more time and effort might well be devoted to the systematic teaching of the theory of relativity than is usual at present at most universities." 4 4 ^(4){ }^{4}4
Years passed, but eventually some charismatic teachers began to heed Einstein's call. Among the first and most influential was John Wheeler, who began to offer Physics 570, a full-length course on general relativity, at Princeton University in 1954. He quickly attracted world-class graduate students to the subject, including Charles Misner and Kip Thorne. Fifteen years later, concerned that textbooks on general relativity had failed to keep up with modern developments, Misner, Thorne, and Wheeler teamed up to write Gravitation. 5 5 ^(5){ }^{5}5 On publication, Gravitation joined several other new books about general relativity, including Steven Weinberg's Gravitation and Cosmology (1972) and Stephen Hawking's and George Ellis's The Large Scale Structure of Space-Time (1973). 6 6 ^(6){ }^{6}6 Unlike those books, however, MTW defied many people's expectations for a textbook. Some just didn't know what to make of it.
Misner, Thorne, and Wheeler clearly intended Gravitation to be a textbook, pitched at advanced physics students. Wheeler's notes from an early planning meeting with his coauthors made clear that they would write the book with "the committee planning graduate courses in U . of X " in mind. While certainly thinking in terms of a textbook, however, from the start they treated the project as an experiment in the genre. The book was organized into two tracks: a core of introductory material occupying less than a third of the book, surrounded by extensions, elaborations, and applications. 7 7 ^(7){ }^{7}7 The two tracks were not sequential; many chapters were divided, section by section, into one track or the other. Even more novel was the extensive use of "boxes" for complementary material. The boxes were set off from the main text by heavy black lines, interrupting the flow of ordinary chapter exposition, often for several pages at a time. Some of the boxes resembled the sidebars that had long been a staple of science textbooks aimed at younger students, featuring short biographies of famous physicists or brief descriptions of important experiments. But most of the boxes in Gravitation served a different purpose. According to Wheeler's notes, the boxes were meant to constitute "a third channel of pedagogy," beyond the two tracks. "They are distinguished from the main text by untidiness" and included "the kinds of things we would like to present in lecture hour to students who can be relied upon to learn tightly organized material and computational methods on their own from a systematic text." Their pedagogical aspirations were clear: as each author drafted a section of the book, the coauthors would "test a write up by asking if a student could use it to lecture from." 8 8 ^(8){ }^{8}8
The authors devoted spectacular attention to the physical appearance and production of the book. Thorne traded detailed letters with the artists and layout designers at the original publisher (W. H. Freeman in San Francisco), going over everything from the thickness of lines to set off the box material to arrow styles and shadings to be adopted in the hundreds of illustrations. Early on, Thorne alerted an editor at Freeman that "several features of the manuscript will require special typesetting problems." Beyond the extensive figures, tables, and boxes, the authors anticipated the need for at least six distinct typefaces, perhaps as many as eight, to properly distinguish the plethora of symbols and equations they would be treating.' (Before the book had even been published, Thorne worried that "the
extreme complexity of the typography" would bedevil foreign-language publishers. He recommended that they simply photograph the equations from the English edition once it became available rather than attempt to retypeset them. 10 10 ^(10){ }^{10}10 ) Given the book's unusual organization, the authors also inserted thousands of marginal comments throughout the book. Some comments summarized the material under discussion, but many others were "dependency statements": a roadmap spelling out which other sections a given discussion depended on, and which others would in turn depend on it. 11 11 ^(11){ }^{11}11
Having tackled every detail of composition and typesetting, imagine the authors' surprise when-two years into the process, and just three weeks before they submitted their final, edited manuscript - they learned that the publisher held a rather different conception of the book than they did. After meeting with their editor from the press, Thorne shot off a letter to his coauthors. "I was rather shocked to learn from Bruce [Armbruster, the editor] that the people at Freeman are so out-of-touch with our book that they have not been regarding it as a textbook, but rather as a technical monograph. I suppose that the enormous size of the book has something to do with it." The publisher's plan had been to produce an expensive hardcover edition, intended primarily for purchase by libraries: "Freeman had not been expecting to pick up the textbook market with this book" at all. Thorne worked hard to convince the editor that "there might be some hope of picking up student sales" as well, but that would require a complete overhaul of the publisher's printing and pricing plans. 12 12 ^(12){ }^{12}12
Was Gravitation a reference monograph for libraries or a textbook for classroom use? From that ontological difference sprang more immediate considerations. For example, how could they keep such a fabulous concoction from crumbling under its own weight? The book's unusual trim size-each of its nearly 1,300 pages was more than an inch wider and taller than standard textbooks at the time - suggested hardcover rather than paperback binding. Hardcover binding seemed all the more appropriate to the authors, for whom Gravitation was self-evidently a textbook, since (as Thorne explained), "it seems to me that paperback editions cannot hold up well enough with the heavy use that a student in a full year course would give the book." But hardcover binding threatened to price the book beyond the reach of a student market. 13 13 ^(13){ }^{13}13 After assurances from the publisher that paperback binding could hold up just as ruggedly as hardcover, the authors struck a deal with W. H. Freeman: in exchange for reduced royalty rates on the paperback edition, the press would aim to keep the price of the paperback lower than the hardcover price of the recent textbook by Weinberg, Gravitation and Cosmology. On publication, the paperback edition of Misner, Thorne, and Wheeler's Gravitation sold for $ 19.95 $ 19.95 $19.95\$ 19.95$19.95 (about $ 110 $ 110 $110\$ 110$110 in 2017 dollars),
and the hardcover for twice that price. With the publisher now treating the book as a textbook rather than a reference monograph, and with the compromise pricing plan in place, Thorne was confident that the book could "capture one hundred percent of the textbook market in this field-or as nearly so as possible." 14 14 ^(14){ }^{14}14
Like the authors and publisher, reviewers recognized the book as unusual. "A pedagogic masterpiece," announced a reviewer in Science; "one of the great books of science, a lamp to illuminate this Aladdin's cave of theoretical physics whose genie was Albert Einstein," crowed another in Science Progress. A third reviewer challenged his readers: "Imagine that three highly inventive people get together to invent a scientific book. Not just to write it, but invent the tone, the style, the methods of exposition, the format." Many reviewers lauded the rich set of illustrations and the innovative use of boxes. 15 15 ^(15){ }^{15}15 Others complained that the two-track-plus-box organization introduced too many redundancies. "This is a difficult book to read in a linear, progressive fashion," concluded one reviewer; "there is needless repetition (indeed, almost everything is stated at least three times)," noted another. "The variety of gimmicks is bewildering - framed headings with quotations, marginal titles, 'boxes' sometimes extending over several pages, heavy type, light type, large type, small type," reported a reviewer in Contemporary Physics. "Clearly the book is an experiment in presentation on a grand scale. 1 / 6 1 / 6 ^(1//6){ }^{1 / 6}1/6
Nearly all reviewers commented on the writing style. Wheeler was already well known among physicists for his catchy slogans and engaging prose. (Among other memorable contributions, Wheeler had coined the term "black hole.") Wheeler's early planning notes for the book insisted that he and his coauthors must "make clear the idea itself. But soberly, factually, no hyperbole, no enthusiasm. 1 17 1 17 ^(1^(17)){ }^{1{ }^{17}}117 If that had been the intention, not all reviewers agreed on the outcome. The book featured a "prose style varying from the unusually colloquial to the unusually lyrical," wrote one reviewer. But one person's lyricism was another's doggerel. "There is a commendable attempt at informality, but this reviewer found the
breeziness irritating at times," came one verdict. "A 'poetical' style is understandable if one deals with such [speculative] topics as 'pregeometry.' However, 'poetical' passages in differential geometry, for example, may obstruct the understanding of an ascetic reader," concluded another. 18 18 ^(18){ }^{18}18 One reviewer huffed that the informal writing style "comes dangerously close to being patronisingly simplistic, to the point of insulting the reader's intelligence." Another reviewer was even more scandalized by the book's tone. The intended reader, he scoffed, would be most at home with the book "if he is a regular subscriber to Time magazine-the writing of these authors has much in common with its breathless style. 19 19 ^(19){ }^{19}19 Subrahmanyan Chandrasekhar, the famed astrophysicist and Nobel laureate who had grown up in India, trained in Britain, and settled in the United States, likewise noted that the book's "style fluctuates from precise mathematical rigor to evangelical rhetoric." He closed his review with the memorable observation: "There is one overriding impression this book leaves. 'It is written with the zeal of a missionary preaching to cannibals' (as J. E. Littlewood, in referring to another book, has said). But I (probably for historical reasons) have always been allergic to missionaries." (Thorne wrote to Chandrasekhar that the closing paragraph had left him "chuckling for about ten minutes." ) 20 ) 20 )^(20))^{20})20
While acknowledging the book's unusual organization, writing style, and pedagogical innovations, most reviewers treated the book as the authors had intended: a textbook primarily for graduate-level coursework in the technical details of gravitational physics. The authors had set out to corner the market for textbooks on the topic, and they largely succeeded. A few years after publication, their book was still selling between 4,000 and 5,000 copies per year, while their main competitor, Weinberg's Gravitation and Cosmology, had dropped to around 1,000 copies per year. Thorne noted to the original publisher-with fanfare but not much hyperbole - that by the late 1970s, "a large fraction of the physics graduate students in the Western world bought a copy of Gravitation. " 21 " 21 ^("21){ }^{" 21}"21 The book sold 50,000 copies during its first decade, at a time when institutions in the United States graduated about 1 , 000 PhDs 1 , 000 PhDs 1,000PhDs1,000 \mathrm{PhDs}1,000PhDs in physics per year, and no other country came close to those annual totals. 22 22 ^(22){ }^{22}22
Yet from the start, some readers saw much more in Gravitation than a vehicle for training soon-to-be specialists. The original publisher, for one, reversed course in a dramatic way. A decade after having written off the book as merely a reference work for library purchase, editors at W. H. Freeman decided to advertise a specially reduced price on the book - nearly 25 % 25 % 25%25 \%25% off list price - to subscribers of the popular magazine Scientific American. Thorne countered that a better way to test "the elasticity in the demand" for the book would be to offer that reduced price to "that portion of the market which concerns me most": students and young academics. He urged the publisher to offer the reduced price to university bookstores rather than Scientific American devotees. 23 23 ^(23){ }^{23}23
Nonetheless, the publisher was on to something. On the book's publication, reviews had run not just in such venues as Science and Physics Today; the Washington Post devoted a full-page review to the book, and a daily newspaper in San Antonio, Texas, likewise recommended it. The reviewer in the Post, himself a physicist at Williams College, acknowledged that "perhaps it is strange to review here a textbook full of mathematics, a book, moreover, whose 6.7 -pound bulk the young, the old and the infirm can scarcely lift. But," he declared, "those who read like to know what is being published and discussed." And Gravitation certainly warranted discussion. The book's engaging prose "awakens hope that the fuzzy and lugubrious 'style' that still spreads its gloom over so much of American science may not be in fashion forever." Moreover, the book's unusual organization seemed akin to recent trends in avant-garde filmmaking, such as the French nouvelle vague. "There are very few stories that should be told sequentially," the reviewer avowed. All the better that Gravitation, like the hip filmmakers, had discovered "strategies for breaking up a linear narrative., 24 24 ^(24){ }^{24}24 The San Antonio reviewer likewise encouraged his readers. "I am not a mathematician, and the 200 or so pages I've read are not all that formidable," he explained. "If you're curious and have an imagination, you won't be cowed. The challenge is stiff, but fascinating." The organization of the book was "phenomenal," and the topic inspiring. He concluded, "This is a fabulous, rewarding book." Novelists could scarcely hope for a more enthusiastic review. 25 25 ^(25){ }^{25}25
Fan letters also streamed in to the authors from a wide assortment of readers. Many came from colleagues who reported how much they enjoyed teaching from the book in their formal classes. 26 26 ^(26){ }^{26}26 But others came from further afield. One reader wrote from a hospital in Italy - it is not clear whether the handwritten letter came from a patient or a physician - to
press the authors on whether their views about the cosmos had changed during the three years since the book's publication. (The letter writer had been keeping up with more recent discussions in the field by reading the Italian-language version of Scientific American.) He had more specific questions, too. In particular, what was the fate of life in a universe that cycled from big bang to big crunch? He was so desperate for a response that he promised $ 200 $ 200 $200\$ 200$200 to anyone (the authors or their graduate students) who might take the time to answer. "Don't be offended by my proposal. Time = = === Money." 27 27 ^(27){ }^{27}27
An engineer in Brussels turned to the book for a different reason. He decided to pick up Gravitation to help him learn English before beginning military service. "My hopes have been completely fulfilled: Gravitation is worth reading to learn English because it makes enjoy Physics!" The book so inspired him that he drew seven full-page, whimsical cartoons in the style of Antoine de Saint-Exupéry's The Little Prince to illustrate concepts he had learned from Gravitation. 28 28 ^(28){ }^{28}28
Readers closer to home wrote to the authors as well. Especially poignant was a letter that Thorne received from a reader in Portland, Oregon. "I stumble here, fall down there, and generally make a fool of myself as I wander about your textbook," the correspondent explained, "but I am gaining a sense of balance and a few tools with which to deal with the subject." His dedication to the book was impressive:
When friends ask me about what I am doing I have made the mistake of telling them the truth [about his attempts to read Gravitation]. Sometimes I think they are right, I feel as though I am on the brink of madness. I go out to have a beer and listen to someone talk about his love affairs, the clutch in his pick-up truck, the problems with his children, the plumbing, the bus service. I look at him and see him dealing with all these important issues and I ask myself why do I care if I ever understand the difference between leptons and leprosy?
Yet still he could not shake his "obsession" with Einstein's own question, "whether or not God had any choice in the creation of the Universe." He needed to know: "Could God be a traveling technician whose responsibility is to supervise gravitational collapses and big bangs? 2 29 2 29 ^(2)^(29){ }^{2}{ }^{29}229
Six years after publication, with annual sales still brisk, John Wheeler tried to assess the reasons for the book's success. Writing to his editor, Wheeler surmised that "many people buy the book who are attracted by the mystique, the boxes, the interesting illustrations, the ideas but who don't expect to and never will get deep into the mathematics." He figured
about half the purchasers fell into that category-and he was eager not to lose them. In thinking about revising and updating the book, Wheeler concluded that "I think we can add a few things and take away a lot of things to keep this group 'on board. 1 30 1 30 ^(1^(30)){ }^{1{ }^{30}}130 Those plans fell through - Misner, Thorne, and Wheeler never did undertake a revision of their massive masterpiece-but Wheeler's observation nonetheless rang true. In their effort to write a specialized textbook, they had produced a hybrid work, as attractive to Scientific American subscribers for its "mystique" as to doctoral students struggling to enter the field.
Gravitation narrowly escaped the pigeonhole of library-only reference work and went on to sell tens of thousands of copies. The book received extensive analysis and review in physicists' specialist journals, even as it inspired passion-even ecstacy-among journalists and nonspecialist readers. Somehow this hulking book, stuffed to overflowing with equations so complicated they required multiple typefaces and elaborate marginal notes, excited broad, crossover appeal. To this day, Misner, Thorne, and Wheeler's book-like Einstein's elegant theory at its core - continues to inspire alluringly large questions. Why are we here? What is our place in the cosmos? Einstein helped spur those questions a century ago. Gravitation marks a major milestone on that continuing quest. It is a tribute to Princeton University Press that this fascinating book-with its jumble of equations, nonlinear structure, and, at times, soaring lyricism - will once again be easily available to students and seekers alike.

ACKNOWLEDGMENTS

My thanks to Professor Kip Thorne for generously sharing his personal files with me and for granting permission to quote from them; to Charles Greifenstein and Anne Harney of the American Philosophical Society for their assistance with the John A. Wheeler papers; and to Marie Burks, Michael Gordin, Yoshiyuki Kikuchi, Roberto Lalli, Bernard Lightman, David Singerman, Alma Steingart, Marga Vicedo, Benjamin Wilson, and Aaron Wright for helpful comments on an earlier draft. I also thank Ingrid Gnerlich for her kind invitation to contribute this Foreword to the new edition of Misner, Thorne, and Wheeler's magnificent volume, Gravitation.

PREFACE TO THE 2017 PRINTING OF GRAVITATION CHARLES W. MISNER AND KIP S. THORNE

As we look back on our sixty-year love affair with Einstein's general relativity, our primary emotion is joy: joy at having participated in an amazingly fruitful era of exploration and transformation. In geologists' terminology, we have lived a blessed bit of the Anthropocene epoch from a favored perch in the world, seeing wonders, while, fortunately, avoiding personally the wars and devastations that have afflicted so many others.
There is an immense contrast in human understanding of gravity in action from the 1950s, when John Wheeler recruited us into Einstein's arena, to the present time. In the 1950s, curved spacetime was a complex though beautiful way to interpret one observational datum from each of four phenomena: the bending of light by the Sun, the perihelion motion of Mercury, the gravitational redshift from the white dwarf 40 Eridani B, and the expansion of the universe. Today we have observational data by the megabyte. The icons for these data are (1) the WMAP-based plot of the variations of temperature of the cosmic microwave radiation as a function of angular scale - the marker for the advent of precision cosmology; and (2) the "chirp" plots of LIGO's first directly observed gravitational wave, marking the advent of gravitational wave astronomy. Along with these icons, there has been a wealth of other great insights and discoveries as the general relativity community expanded from a few dozen to a few thousand during the six decades since 1952, when John Wheeler began dreaming of this textbook.

the context in which we wrote gravitation

General relativity had an exciting first two decades (1915-1939) and then became a twodecade backwater for physicists (1939-1958), as nuclear physics, elementary particle physics, and condensed matter physics came to the fore. In parallel, in mathematics, the field now called differential geometry was blossoming. For example, the concept of a manifold
was clarified in the decades of the 1930s through the 1950s, and Milnor (1956) placed a capstone on this progress when he showed, by an example with the seven-dimensional sphere, that two manifolds that are equivalent at the level of continuous functions could be different in an essential way at the level of differentiable functions.
We were fortunate to enter relativity near the beginning of a remarkable renaissance (ca. 1958-1978), one enabled in part by the new mathematics and driven initially, in large measure, by our mentor John Wheeler, and then driven by a sequence of astronomical discoveries: the cosmic microwave background (CMB), and phenomena associated with black holes and neutron stars: quasars, pulsars, jets from galactic nuclei, compact X-ray sources, and gamma-ray bursts. It was late in this renaissance that we wrote Gravitation.
The relativity textbooks that preceded Gravitation were too old to incorporate the wonderful new observations and the new mathematical underpinnings. They treated Riemannian geometry as Einstein and then Pauli (1921) had, with almost no concept of the idea of a topological manifold that could carry properties (such as tangent vectors and 1 -forms) even though it had been assigned no metric. They also, then, used no idea of points in the manifold (events in spacetime) as being conceptually superior to the various lists of coordinates used to identify them. And these texts tended to describe the physics almost entirely in terms of (old-fashioned) mathematics, with little attention paid to the heuristic but powerful tools by which modern physicists make rapid progress: physical arguments and pictures, geometric diagrams, and intuitive viewpoints. In Gravitation, our goal was to present relativity in physicists' physical, visual, and intuitive language, accompanied by the modern mathematics from which this language springs. The result was an advanced textbook with a far larger word-to-equation ratio than anything ever before seen in this field; a book filled with "purple prose," as John's wife, Janette, referred to it. But a book that also teaches relativity's mathematical underpinnings.
With our purple prose and pictures, we sought to transform how scientists think about relativity. And we think we succeeded, at least to some degree.

GRAVITATION'S GEOMETRIC VIEWPOINT

A major part of our approach is the geometric viewpoint on general relativity that we learned from John-a viewpoint that contrasts starkly with the field-theoretic viewpoint taken by Steven Weinberg in the relativity textbook (Weinberg 1972) that he wrote in parallel with our writing Gravitation.
For situations where spacetime is strongly curved and where we focus on regions comparable to or larger than its radius of curvature (e.g., black holes and a closed model universe), this geometric viewpoint is essential, or at least superior. For the causal structure of spacetime (horizons, singularities, Hawking's second law of black hole mechanics), it is also essential. For most other situations, while not essential, it is powerful. And whenever field-theoretic techniques are more useful than geometry (e.g., in the evolution of structure
in the early universe), one can easily descend from the heights of geometry to the nitty gritty of field theory. (OK. Our prejudice is showing. Starkly.)
After decades steeped in the geometric viewpoint, one of us (Kip) has become so enamored of it that, with Stanford astrophysicist Roger Blandford, he has crafted a much broader textbook permeated with this viewpoint: a book titled Modern Classical Physics (Thorne and Blandford 2017; henceforth "MCP"), which covers all the areas of classical physics that PhD physicists should be exposed to but often are not, at least in North America. That book and this reprinting of Gravitation are being published simultaneously by the same publisher, Princeton University Press.

HOW USEFUL CAN GRAVITATION BE TODAY?

Gravitation was published in 1973, near enough to the end of the Relativistic Renaissance that most of that Renaissance's major theoretical insights and observational discoveries were in hand. While there have been some major additional insights and discoveries in the four decades since, they are few enough that Gravitation is seriously out of a date in only a moderate number of areas; primarily cosmology (Part VI), gravitational waves (Part VIII), experimental tests of general relativity (Part IX), and observations but not the theory of black holes and neutron stars (Parts V and VII).
This may account, in part, for Gravitation's longevity: it continues to be used as supplemental reading in a large number of relativity courses around the world even today, 44 years after its publication. And in recent years, it has still been the primary textbook for a few courses.

CHAPTER-BY-CHAPTER STATUS OF GRAVITATION

As an aid to students, teachers, and other readers as they choose a path through relativity in the modern era, we offer here a chapter-by-chapter description of what in Gravitation is out of date and what is not; what is missing that we think so important that we would include it in a full year, advanced course in general relativity if we were teaching one; and where readers can go to learn about the missing developments.
  1. Parts I, II, III, and IV, the fundamentals of general relativity, have not changed significantly over the past 44 years, so Chapters 1-22 on the fundamentals are almost fully up to date. The only exceptions are the following.
    A. Chapter 8, Differential Geometry, should be augmented by an introduction to symbolic manipulation software (e.g., Maple, Mathematica, and Matlab) for computing connection coefficients and curvature tensors and performing other tensorial calculations; and Chapter 14, Calculation of Curvature, could be augmented by a deeper treatment of symbolic manipulation.
    B. To Part IV, Einstein's Geometric Theory of Gravity, we would add four new topics:
    a. Numerical relativity, which underpins gravitational wave observations and is teaching us about the nonlinear dynamics of curved spacetime; for example, Maggiore (2017), or for far greater detail, Baumgarte and Shapiro (2010) and Shibata (2016).
    b. Gravitational lensing, which is based on the linearized approximation to general relativity (Section 18.1) and has become a major tool for astronomy; for example, MCP or Straumann (2013), or for far greater detail, Schneider, Ehlers, and Falco (1992).
    c. The Einstein field equation in higher dimensions, particularly four space dimensions and one time dimension, which is motivated by string theory's requirement for higher dimensions and by the Randall-Sundrum (1999a,b) insight that one or more of these higher dimensions could be macroscopic. This topic often goes under the name "Braneworlds." For a brief treatment see, for example, Zee (2013); for much greater detail at the level of Gravitation, see Maartens and Koyama (2010).
    d. Quantum field theory in curved spacetime (which could be added at the end of Chapter 22). This topic underpins, most importantly, Hawking radiation from black holes; see below. For a brief introduction see, for example, Carroll (2004); for more thorough treatments, see Wald (1994) and Parker and Toms (2009).
  2. Part V, Relativistic Stars, is similarly almost fully up to date, with the following two exceptions.
    A. Chapter 24, Pulsars and Neutron Stars; Quasars and Supermassive Stars, is completely out of date. Observations and observation-driven astrophysical theory have transformed our understanding profoundly. See, for example, Straumann (2013) or Maggiore (2017) or, for far greater detail, Shapiro and Teukolsky (1983), which is somewhat out of date but excellent and thorough.
    B. Chapter 25 on geodesic orbits in the Schwarzschild spacetime should be augmented by exercises on computing orbits numerically to give the reader physical insight-which is best done by numerically integrating the Hamilton equations that follow from the super-Hamiltonian (Exercise 25.2); see Levin and Perez-Giz (2008).
  3. Part VI, The Universe, is for the most part tremendously out of date.
    A. Chapter 27, Idealized Cosmologies, is an exception. The fundamental ideas and equations for idealized cosmologies have not changed, but the emphasis of this chapter is archaic. John, our mentor-whose intuition and prescience
    were usually superb (Misner, Thorne, and Zurek 2009)-was firmly convinced that our universe would turn out to be closed and have vanishing cosmological constant; so in Gravitation, the closed Friedman cosmology is given great emphasis. Since Gravitation was published, a rich set of cosmological observations has revealed that our universe is very nearly flat spatially and has a positive cosmological constant (or something resembling it). So this chapter should be augmented by a more detailed treatment of the material in Section 27.11, and most importantly, by an in-depth treatment of the de Sitter solution of the Einstein equation with cosmological constant - as, for example, in Hawking and Ellis (1973). As a side issue (a Box), we would add the anti-de Sitter (AdS) solution (e.g., Hawking and Ellis 1973), because of its importance today in explorations of fundamental physics (e.g., the AdS/CFT correspondence).
    B. Chapter 28, Evolution of the Universe into Its Present State, and Chapter 29, Present State and Future Evolution of the Universe, are completely out of date and thus only of historic interest. During the past two decades, these subjects have been thoroughly transformed by cosmological observations and associated theory. For a fully up-to-date, pedagogical treatment, we recommend chapter 28 of MCP, or at a more elementary level, Schneider (2015). The most useful advanced textbook may be Weinberg (2008).
    C. Cosmological observations over the past two decades suggest that Chapter 30, Anisotropic and Homogeneous Cosmologies is likely not relevant to the early evolution of our universe. However, it is of great importance for a fundamental new topic to be discussed below (see 4.D): the physical structure of singularities.
    D. To this cosmological Part of Gravitation, we would add a major new topic (chapter): Inflationary expansion in the very early universe, as treated, for example, in Peacock (1999); Hobson, Efstathiou, and Lasenby (2006); Sasaki (2015); and section 28.7.1 of MCP.
  4. Part VII, Gravitational Collapse and Black Holes, is surprisingly up to date, in large measure because it focuses on theory and says little about observations. However, a few new theoretical developments (some major) have emerged since 1973 that should be included in any year-long advanced course on general relativity.
    A. To Chapter 31, Schwarzschild Geometry, and Chapter 32, Gravitational Collapse, we would add nothing.
    B. To Chapter 33, Black Holes, we would add the following topics.
    a. Exercises to explore geodesic orbits around a Kerr black hole numerically, by integrating Hamilton's equations for the super-Hamiltonian (33.27c) (Levin and Perez-Giz 2008).
    b. A discussion of quasinormal modes of a Kerr black hole, motivated by Exercise 33.14; see, for example, chapter 12 of Maggiore (2017). (The first hint of these modes was found by Vishveshwara, 1970, in the form of ringdown waves like those that LIGO has detected 45 years later. By 1973, when Gravitation was published, the concept of quasinormal modes was fully in hand along with the equations for computing them, but the first numerical computation of their complex eigenfrequencies and eigenfunctions, by Chandrasekhar and Detweiler, 1975, was still two years in the future.)
    c. Spherical accretion onto a Schwarzschild black hole and accretion disks around a Kerr black hole: at least a few exercises as, for example, in MCP. These topics are touched on in Box 33.3 of Gravitation, but given their great astrophysical importance today, they deserve greater and more up-to-date detail; see, for example, the brief discussion in Straumann (2013), the longer discussion in Abramovicz and Fragile (2013), or the very detailed discussion in Meier (2012).
    d. The Blandford-Znajek (1977) mechanism by which magnetic fields extract spin energy from black holes to power jets; see, for example, MCP. For far greater detail, see Thorne, Price, and MacDonald (1986), which emphasizes the relativity, and McKinney, Tchekhovskoy, and Blandford (2012), which emphasizes the astrophysics.
    e. Some discussion of astronomical observations of black holes and their astrophysical roles in the universe; see, for example, Narayan and McClintock (2015) or for greater detail, Meier (2012) and Schneider (2015). (What remarkable developments there have been here, since Gravitation was published!)
    f. Hawking radiation, the associated thermal atmosphere of a black hole, and black-hole thermodynamics (all of which were developed within a year of publication of Gravitation, in the wake of Stephen Hawking's and others' bringing quantum field theory in curved spacetime into a sufficiently mature form; see 1.B.c above). See, for example, Carroll (2004) for a moderately brief, pedagogical treatment, and Wald (1994) for greater detail.
    C. To Chapter 34, Global Techniques, Horizons, and Singularity Theorems, we would add a new set of topics that have been explored using global techniques since Gravitation was published: wormholes and topological censorship; and closed timelike curves, chronology horizons, and chronology protection. See, for example, Everitt and Roman (2012) for a not very technical discussion with references to the most important literature or Friedman and Higuchi (2008) for greater technical detail.
    D. To Part VII we would also add the equivalent of one more chapter on The Physical Structure of Generic Singularities and the Interiors of Black Holes. This chapter would include the following.
    a. The material in Chapter 30 (3.C above) on the Kasner and Mixmaster solutions of Einstein's equation, plus a more detailed discussion of the Belinsky, Khalatnikov, and Lifshitz (BKL) analysis, which suggests there is a generic, spatially inhomogeneous variant of Mixmaster (p. 806 of Gravitation); also, a description of numerical relativity simulations (Garfinkle 2004; Lim et al. 2009) that prove this to be true and reveal some surprising twists missed by BKL.
    b. Analyses that show that the inner horizons, r = r , r = r r=r_(", ")r=r_{\text {, }}r=r, of a Kerr or ReissnerNordstrøm black hole (Fig. 34.4) are highly unstable and that material or radiation falling into the hole triggers these instabilities, converting the inner horizons into generic null singularities (Poisson and Israel 1990; Marolf and Ori 2012).
  5. In Part VIII, Gravitational Waves, the chapters on the theory of the waves and their generation are largely up to date, but the chapter on their detection is extremely out of date. More specifically:
    A. Chapter 35, Propagation of Gravitational Waves, is essentially up to date.
    B. The topics covered in Chapter 36, Generation of Gravitational Waves, are essentially up to date, but they need to be augmented by the following.
    a. An overview of gravitational wave sources that are likely to be observed in the next decade or two; see, for example, Buonanno and Sathyaprakash (2015); Creighton and Anderson (2011); or, for far greater detail, Maggiore (2017).
    b. A sketch of the post-Newtonian expansion of the waves from compact binary stars (higher-order corrections to this chapter's quadrupolar analysis); see, for example, Straumann (2013); and for greater detail, Poisson and Will (2014) or Blanchet (2014). Exercise 39.15 of Gravitation could be a starting point for this.
    c. A description of numerical relativity simulations of the inspiral and merger of black-hole binaries, and black-hole/neutron-star binaries, and their gravitational waves (e.g., Choptuik, Lehner, and Pretorius 2015; Maggiore 2017); also, the nonlinear dynamics of curved spacetime triggered by black-hole mergers (e.g., Owen et al. 2011; Scheel and Thorne 2013).
    d. A sketch of the analysis that shows that early-universe inflation parametrically amplifies gravitational vacuum fluctuations coming off the big bang, to produce a spectrum of primordial gravitational waves; see, for example, Mukhanov (2005) or Maggiore (2017).
    C. Chapter 37, Detection of Gravitational Waves, is highly out of date. Although nothing is wrong with this chapter, and it can be of conceptual value (particularly Sections 37.1-37.3), it focuses on vibrating mechanical detectors, which have largely been abandoned. So in a modern course, we would replace Sections 37.4-37.10 by the following.
    a. An overview of the four types of detectors that are expected to open up four different gravitational-wave frequency bands in the next two decades: ground-based interferometers, such as LIGO, which have already opened the high-frequency band ( 10 10 , 000 Hz 10 10 , 000 Hz 10-10,000Hz10-10,000 \mathrm{~Hz}1010,000 Hz ); space-based detectors, such as LISA, in which drag-free spacecraft track each other with laser beams, that are expected to open up the low-frequency band (periods of minutes to hours) in the next 15 or 20 years; pulsar timing arrays (PTAs), which are expected to open the very low frequency band (periods of a year to a few tens of years) in the coming decade; and socalled B-mode polarization patterns in the CMB, which are induced by primordial gravitational waves with periods of millions to billions of years (the extremely low frequency band) and which may be definitively measured in the next decade or so. See, for example, Berger et al. (2015) and Maggiore (2017).
    b. Detailed analyses of (idealized) ground-based interferometers, spacebased detectors, and PTAs (e.g., Creighton and Anderson 2011; Saulson 2017; MCP); and analyses of the influence of gravitational waves on CMB polarization (e.g., Maggiore 2017).
    c. A summary of observations of gravitational waves, which in 2017 are solely those by LIGO, such as Abbott et al. (2016a).
    D. This could be a good spot, in an advanced course, to present an overview of what is known about the nonlinear dynamics of vacuum, curved spacetime (e.g., Scheel and Thorne 2014)-most of which has already been mentioned above.
    a. The chaotic spacetime dynamics near a generic Mixmaster (BKL) singularity (4.D.a).
    b. The more gentle dynamics near a generic null singularity (4.D.b).
    c. The phase transitions, critical behavior, and scaling that show up in (nongeneric) "critical" gravitational collapse; for example, Choptuik, Lehner, and Pretorius (2015).
    d. The interacting "tidal tendices" and "frame-drag vortices" that generate the gravitational waves in black-hole collisions; for example, MCP, or for greater detail Scheel and Thorne (2014), or for still greater detail Owen et al. (2011).
    e. Nonlinear, two-dimensional turbulence (energy cascades from small scales to large scales) triggered by mode-mode coupling in perturbations of a fast spinning black hole; see Yang, Zimmerman, and Lehner (2015).
We suspect that these just "scratch the surface" on nonlinear spacetime dynamics, and that a rich range of other phenomena will be discovered in the coming years.
6. Part IX, Experimental Tests of General Relativity, is all correct, but since Gravitation was published, rapidly improving technology and vigorous efforts by creative experimenters have moved the most accurate experimental tests from errors of a few percent to errors as small as one part in 100,000 ; so, obviously, a huge amount of updating is necessary.
A. A modern course might simply follow the discussion of experimental tests in recent pedagogical references, such as Will ( 2014 , 2015 ) ( 2014 , 2015 ) (2014,2015)(2014,2015)(2014,2015).
B. Or it might do the following.
a. Preserve the discussion of foundational tests in Chapter 38, augmented by an overview of the current status of those and related experiments from Will ( 2014 , 2015 ) ( 2014 , 2015 ) (2014,2015)(2014,2015)(2014,2015).
b. Preserve the pedagogical discussion of the post-Newtonian approximation and the parametrized post-Newtonian formalism in Chapter 39, augmented by the corresponding analysis for the orbital motion of compact binaries (a straightforward extension of Exercise 39.15).
c. Preserve the analysis of solar system experiments in Chapter 40, augmented by an overview of the current status of those experiments as in Will ( 2014 , 2015 ) ( 2014 , 2015 ) (2014,2015)(2014,2015)(2014,2015).
d. Add discussion and some analyses of experimental tests in binary pulsars (e.g., Straumann 2013; Will 2014, 2015); and also experimental tests based on gravitational wave observations of binary black holes, for which expectations are discussed in Yunes and Siemens (2013) and in Gair et al. (2013), and results are just beginning to emerge from LIGO (e.g., Abbott et al. 2016b).
7. Part X, Frontiers, is a beautiful overview of some important ideas that occupied John Wheeler's attention in the era when we wrote this book with him.
A. Chapter 41, Spinors, is an introduction to this important topic in mathematical physics - an introduction that mixes the deep mathematics with the intuitive, visual, and physical viewpoint that was John's hallmark. This chapter stands on its own, with no need for change.
B. The Regge Calculus, laid out so beautifully in Chapter 42, has played a powerful conceptual role in general relativity for decades, but has never (yet) become an effective tool for numerical computations.
C. Superspace, as treated in Chapter 43 , has long been a powerful underpinning for some approaches to formulating laws of quantum gravity.
D. Chapter 44, Beyond the End of Time, describes prescient ideas on which John focused in the 1960s-1980s. It is of great historical import, and it contains ideas that continue to have influence.
We commend these chapters to readers, followed by a perusal of modern applications on the physics archive, https://arxiv.org.
Gravitation and these updates clearly constitute far more material than can be covered in a full year course, just as Gravitation by itself did in 1973, when first published. Today, as then, a teacher or student or reader will want to select which portions to focus on, and at what depth. But the above summary does convey what we think important and worthy of study in 2017.

ACKNOWLEDGMENTS

Above all, we are indebted to our mentor and coauthor, John Archibald Wheeler, who enticed us into the arena of general relativity six decades ago with his optimism, enthusiasm, and eagerness for adventure.
Many colleagues, friends, students-and students of students of students-have rewarded us by embracing this heavy tome. They have our sincere thanks. We hope they continue to appreciate the beauty of the ideas described in our book: the intellectual universe that Einstein opened for humanity more than 100 years ago. And we hope and expect that they will help others, in coming decades, to see the magnificence and subtlety of Nature through this window.
We relied on textbooks, as well as on John, as we struggled to learn general relativity in the 1950s and early 1960s. We thank the authors of those texts: Peter Bergmann, Christian Møller, Richard Tolman, John Synge, and Lev Landau and Evgeny Lifshitz. Beyond these, John Wheeler encouraged Misner to look at the 1955 text by André Lichnerowicz, which does reflect the then-current differential geometry. Much of Part III in Gravitation reflects Misner's efforts to help Wheeler restate the differential geometry that Misner had learned from his Notre Dame mathematics mentor Arnold Ross, his Princeton mathematics advisor Donald Spencer, and fellow Princeton graduate students.
For the research that has transformed this field, financial support was needed. On behalf of our colleagues as well as ourselves, we thank the program directors who targeted that support with great wisdom over the early years, particularly Joshua Goldberg (for the U.S. Air Force), and Harry Zapolsky and then Richard Isaacson (for the National Science
Foundation), followed by many others when the necessary investments got large. The impact of gravitational wave observations will be huge over the coming decades. The chief architect on the rocky course from small-scale R&D to the massively big collaborations required for success was Barry Barish, while Joseph Weber's pioneering insights, courage, and determination from his beginnings with Wheeler in 1956 should never be overlooked.
For helping bring this textbook to fruition in 1973, we are indebted to many people; see the original Acknowledgments on page li.
Until 2015, Gravitation continued to sell many hundred copies a year. Then, through a series of acquisitions of publishers by publishers by publishers, it wound up in the hands of Macmillan, which took it out of print. Through persistence, finesse, and firmness, Joan Winstein succeeded in extracting all rights to Gravitation from Macmillan, and to her we are deeply grateful.
We were fortunate that Princeton University Press eagerly embraced the idea of producing a new hardback printing at a remarkably low purchase price. We thank the superb staff at the Press, who have worked so effectively with us to bring this printing to fruition, particularly Peter Dougherty, Ingrid Gnerlich, Arthur Werneck, Karen Carter, Lisa Black, and Jessica Massabrook.
And we thank David Kaiser for his beautiful new Foreword to this printing.
July 1, 2017

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PREFACE

This is a textbook on gravitation physics (Einstein's "general relativity" or "geometrodynamics"). It supplies two tracks through the subject. The first track is focused on the key physical ideas. It assumes, as mathematical prerequisite, only vector analysis and simple partial-differential equations. It is suitable for a one-semester course at the junior or senior level or in graduate school; and it constitutes-in the opinion of the authors-the indispensable core of gravitation theory that every advanced student of physics should learn. The Track-1 material is contained in those pages of the book that have a 1 outlined in gray in the upper outside corner, by which the eye of the reader can quickly pick out the Track-1 sections. In the contents, the same purpose is served by a gray bar beside the section, box, or figure number.
The rest of the text builds up Track 1 into Track 2. Readers and teachers are invited to select, as enrichment material, those portions of Track 2 that interest them most. With a few exceptions, any Track-2 chapter can be understood by readers who have studied only the earlier Track-1 material. The exceptions are spelled out explicitly in "dependency statements" located at the beginning of each Track-2 chapter, or at each transition within a chapter from Track 1 to Track 2.
The entire book (all of Track 1 plus all of Track 2) is designed for a rigorous, full-year course at the graduate level, though many teachers of a full-year course may prefer a more leisurely pace that omits some of the Track-2 material. The full book is intended to give a competence in gravitation physics comparable to that which the average Ph.D. has in electromagnetism. When the student achieves this competence, he knows the laws of physics in flat spacetime (Chapters 1-7). He can predict orders of magnitude. He can also calculate using the principal tools of modern differential geometry (Chapters 8-15), and he can predict at all relevant levels of precision. He understands Einstein's geometric framework for physics (Chapters
16-22). He knows the applications of greatest present-day interest: pulsars and neutron stars (Chapters 23-26); cosmology (Chapters 27-30); the Schwarzschild geometry and gravitational collapse (Chapters 31-34); and gravitational waves (Chapters 35-37). He has probed the experimental tests of Einstein's theory (Chapters 38-40). He will be able to read the modern mathematical literature on differential geometry, and also the latest papers in the physics and astrophysics journals about geometrodynamics and its applications. If he wishes to go beyond the field equations, the four major applications, and the tests, he will find at the end of the book (Chapters 41-44) a brief survey of several advanced topics in general relativity. Among the topics touched on here, superspace and quantum geometrodynamics receive special attention. These chapters identify some of the outstanding physical issues and lines of investigation being pursued today.
Whether the department is physics or astrophysics or mathematics, more students than ever ask for more about general relativity than mere conversation. They want to hear its principal theses clearly stated. They want to know how to "work the handles of its information pump" themselves. More universities than ever respond with a serious course in Einstein's standard 1915 geometrodynamics. What a contrast to Maxwell's standard 1864 electrodynamics! In 1897, when Einstein was a student at Zurich, this subject was not on the instructional calendar of even half the universities of Europe. 1 1 ^(1){ }^{1}1 "We waited in vain for an exposition of Maxwell's theory," says one of Einstein's classmates. "Above all it was Einstein who was disappointed," 2 2 ^(2){ }^{2}2 for he rated electrodynamics as "the most fascinating subject at the time" 3 3 ^(3){ }^{3}3-as many students rate Einstein's theory today!
Maxwell's theory recalls Einstein's theory in the time it took to win acceptance. Even as late as 1904 a book could appear by so great an investigator as William Thomson, Lord Kelvin, with the words, "The so-called 'electromagnetic theory of light' has not helped us hitherto . . . it seems to me that it is rather a backward step . . . the one thing about it that seems intelligible to me, I do not think is admissible . . . that there should be an electric displacement perpendicular to the line of propagation." 4 4 ^(4){ }^{4}4 Did the pioneer of the Atlantic cable in the end contribute so richly to Maxwell electrodynamics-from units, and principles of measurement, to the theory of waves guided by wires-because of his own early difficulties with the subject? Then there is hope for many who study Einstein's geometrodynamics today! By the 1920's the weight of developments, from Kelvin's cable to Marconi's wireless, from the atom of Rutherford and Bohr to the new technology of highfrequency circuits, had produced general conviction that Maxwell was right. Doubt dwindled. Confidence led to applications, and applications led to confidence.
Many were slow to take up general relativity in the beginning because it seemed to be poor in applications. Einstein's theory attracts the interest of many today because it is rich in applications. No longer is attention confined to three famous but meager tests: the gravitational red shift, the bending of light by the sun, and
the precession of the perihelion of Mercury around the sun. The combination of radar ranging and general relativity is, step by step, transforming the solar-system celestial mechanics of an older generation to a new subject, with a new level of precision, new kinds of effects, and a new outlook. Pulsars, discovered in 1968, find no acceptable explanation except as the neutron stars predicted in 1934, objects with a central density so high ( 10 14 g / cm 3 ) 10 14 g / cm 3 (∼10^(14)(g)//cm^(3))\left(\sim 10^{14} \mathrm{~g} / \mathrm{cm}^{3}\right)(1014 g/cm3) that the Einstein predictions of mass differ from the Newtonian predictions by 10 to 100 per cent. About further density increase and a final continued gravitational collapse, Newtonian theory is silent. In contrast, Einstein's standard 1915 geometrodynamics predicted in 1939 the properties of a completely collapsed object, a "frozen star" or "black hole." By 1966 detailed digital calculations were available describing the formation of such an object in the collapse of a star with a white-dwarf core. Today hope to discover the first black hole is not least among the forces propelling more than one research: How does rotation influence the properties of a black hole? What kind of pulse of gravitational radiation comes off when such an object is formed? What spectrum of x -rays emerges when gas from a companion star piles up on its way into a black hole? 5 5 ^(5){ }^{5}5 All such investigations and more base themselves on Schwarzschild's standard 1916 static and spherically symmetric solution of Einstein's field equations, first really understood in the modern sense in 1960, and in 1963 generalized to a black hole endowed with angular momentum.
Beyond solar-system tests and applications of relativity, beyond pulsars, neutron stars, and black holes, beyond geometrostatics (compare electrostatics!) and stationary geometries (compare the magnetic field set up by a steady current!) lies geometrodynamics in the full sense of the word (compare electrodynamics!). Nowhere does Einstein's great conception stand out more clearly than here, that the geometry of space is a new physical entity, with degrees of freedom and a dynamics of its own. Deformations in the geometry of space, he predicted in 1918, can transport energy from place to place. Today, thanks to the initiative of Joseph Weber, detectors of such gravitational radiation have been constructed and exploited to give upper limits to the flux of energy streaming past the earth at selected frequencies. Never before has one realized from how many kinds of processes significant gravitational radiation can be anticipated. Never before has there been more interest in picking up this new kind of signal and using it to diagnose faraway events. Never before has there been such a drive in more than one laboratory to raise instrumental sensitivity until gravitational radiation becomes a workaday new window on the universe.
The expansion of the universe is the greatest of all tests of Einstein's geometrodynamics, and cosmology the greatest of all applications. Making a prediction too fantastic for its author to credit, the theory forecast the expansion years before it was observed (1929). Violating the short time-scale that Hubble gave for the expansion, and in the face of "theories" ("steady state"; "continuous creation") manufactured to welcome and utilize this short time-scale, standard general relativity resolutely persisted in the prediction of a long time-scale, decades before the astro-
physical discovery (1952) that the Hubble scale of distances and times was wrong, and had to be stretched by a factor of more than five. Disagreeing by a factor of the order of thirty with the average density of mass-energy in the universe deduced from astrophysical evidence as recently as 1958, Einstein's theory now as in the past argues for the higher density, proclaims "the mystery of the missing matter," and encourages astrophysics in a continuing search that year by year turns up new indications of matter in the space between the galaxies. General relativity forecast the primordial cosmic fireball radiation, and even an approximate value for its present temperature, seventeen years before the radiation was discovered. This radiation brings information about the universe when it had a thousand times smaller linear dimensions, and a billion times smaller volume, than it does today. Quasistellar objects, discovered in 1963, supply more detailed information from a more recent era, when the universe had a quarter to half its present linear dimensions. Telling about a stage in the evolution of galaxies and the universe reachable in no other way, these objects are more than beacons to light up the far away and long ago. They put out energy at a rate unparalleled anywhere else in the universe. They eject matter with a surprising directivity. They show a puzzling variation with time, different between the microwave and the visible part of the spectrum. Quasistellar objects on a great scale, and galactic nuclei nearer at hand on a smaller scale, voice a challenge to general relativity: help clear up these mysteries!
If its wealth of applications attracts many young astrophysicists to the study of Einstein's geometrodynamics, the same attraction draws those in the world of physics who are concerned with physical cosmology, experimental general relativity, gravitational radiation, and the properties of objects made out of superdense matter. Of quite another motive for study of the subject, to contemplate Einstein's inspiring vision of geometry as the machinery of physics, we shall say nothing here because it speaks out, we hope, in every chapter of this book.
Why a new book? The new applications of general relativity, with their extraordinary physical interest, outdate excellent textbooks of an earlier era, among them even that great treatise on the subject written by Wolfgang Pauli at the age of twenty-one. In addition, differential geometry has undergone a transformation of outlook that isolates the student who is confined in his training to the traditional tensor calculus of the earlier texts. For him it is difficult or impossible either to read the writings of his up-to-date mathematical colleague or to explain the mathematical content of his physical problem to that friendly source of help. We have not seen any way to meet our responsibilities to our students at our three institutions except by a new exposition, aimed at establishing a solid competence in the subject, contemporary in its mathematics, oriented to the physical and astrophysical applications of greatest present-day interest, and animated by belief in the beauty and simplicity of nature.

High IslandSouth Bristol, MaineSeptember 4, 1972

Charles W. Misner
Kip S. Thorne
John Archibald Wheeler

ACKNOWLEDGMENTS

Deep appreciation goes to all who made this book possible. A colleague gives us a special lecture so that we may adapt it into one of the chapters of this book. Another investigator clears up for us the tangled history of the production of matter out of the vacuum by strong tidal gravitational forces. A distant colleague telephones in references on the absence of any change in physical constants with time. One student provides a problem on the energy density of a null electromagnetic field. Another supplies curves for effective potential as a function of distance. A librarian writes abroad to get us an article in an obscure publication. A secretary who cares types the third revision of a chapter. Editor and illustrator imaginatively solve a puzzling problem of presentation. Repeat in imagination such instances of warm helpfulness and happy good colleagueship times beyond count. Then one has some impression of the immense debt we owe to over a hundred-fifty colleagues. Each face is etched in our mind, and to each our gratitude is heartfelt. Warm thanks we give also to the California Institute of Technology, the Dublin Institute for Advanced Studies, the Institute for Advanced Study at Princeton, Kyoto University, the University of Maryland, Princeton University, and the University of Texas at Austin for hospitality during the writing of this book. We are grateful to the Academy of Sciences of the U.S.S.R., to Moscow University, and to our Soviet colleagues for their hospitality and the opportunity to become better acquainted in June-July 1971 with Soviet work in gravitation physics. For assistance in the research that went into this book we thank the National Science Foundation for grants (GP27304 and 28027 to Caltech; GP17673 and GP8560 to Maryland; and GP3974 and GP7669 to Princeton); the U.S. Air Force Office of Scientific Research (grant AF49-638-1545 to Princeton); the U.S. National Aeronautics and Space Agency (grant NGR 05-002-256 to Caltech, NSG 210-002-010 to Maryland); the Alfred P. Sloan Foundation for a fellowship awarded to one of us (K.S.T.); and the John Simon Guggenheim Memorial Foundation and All Souls College, Oxford, England, for fellowships awarded to another of us (C.W.M.).
PART

SPACETIME PHYSICS

Wherein the reader is led, once quickly (§1.1), then again more slowly, down the highways and a few byways of Einstein's geometrodynamicswithout benefit of a good mathematical compass.

CHAPTER

GEOMETRODYNAMICS IN BRIEF

§1.1. THE PARABLE OF THE APPLE

One day in the year 1666 Newton had gone to the country, and seeing the fall of an apple, as his niece told me, let himself be led into a deep meditation on the cause which thus draws every object along a line whose extension would pass almost through the center of the Earth.
VOLTAIRE (1738)
Once upon a time a student lay in a garden under an apple tree reflecting on the difference between Einstein's and Newton's views about gravity. He was startled by the fall of an apple nearby. As he looked at the apple, he noticed ants beginning to run along its surface (Figure 1.1). His curiosity aroused, he thought to investigate the principles of navigation followed by an ant. With his magnifying glass, he noted one track carefully, and, taking his knife, made a cut in the apple skin one mm above the track and another cut one mm below it. He peeled off the resulting little highway of skin and laid it out on the face of his book. The track ran as straight as a laser beam along this highway. No more economical path could the ant have found to cover the ten cm from start to end of that strip of skin. Any zigs and zags or even any smooth bend in the path on its way along the apple peel from starting point to end point would have increased its length.
"What a beautiful geodesic," the student commented.
His eye fell on two ants starting off from a common point P P PPP in slightly different directions. Their routes happened to carry them through the region of the dimple at the top of the apple, one on each side of it. Each ant conscientiously pursued
Figure 1.1.
The Riemannian geometry of the spacetime of general relativity is here symbolized by the two-dimensional geometry of the surface of an apple. The geodesic tracks followed by the ants on the apple's surface symbolize the world line followed through spacetime by a free particle. In any sufficiently localized region of spacetime, the geometry can be idealized as flat, as symbolized on the apple's two-dimensional surface by the straight-line course of the tracks viewed in the magnifying glass ("local Lorentz character" of geometry of spacetime). In a region of greater extension, the curvature of the manifold (four-dimensional spacetime in the case of the real physical world; curved two-dimensional geometry in the case of the apple) makes itself felt. Two tracks Q Q Q\mathscr{Q}Q and B B B\mathscr{B}B, originally diverging from a common point P P P\mathscr{P}P, later approach, cross, and go off in very different directions. In Newtonian theory this effect is ascribed to gravitation acting at a distance from a center of attraction, symbolized here by the stem of the apple. According to Einstein a particle gets its moving orders locally, from the geometry of spacetime right where it is. Its instructions are simple: to follow the straightest possible track (geodesic). Physics is as simple as it could be locally. Only because spacetime is curved in the large do the tracks cross. Geometrodynamics, in brief, is a double story of the effect of geometry on matter (causing originally divergent geodesics to cross) and the effect of matter on geometry (bending of spacetime initiated by concentration of mass, symbolized by effect of stem on nearby surface of apple).
his geodesic. Each went as straight on his strip of appleskin as he possibly could. Yet because of the curvature of the dimple itself, the two tracks not only crossed but emerged in very different directions.
"What happier illustration of Einstein's geometric theory of gravity could one possibly ask?" murmured the student. "The ants move as if they were attracted by the apple stem. One might have believed in a Newtonian force at a distance. Yet from nowhere does an ant get his moving orders except from the local geometry along his track. This is surely Einstein's concept that all physics takes place by 'local action.' What a difference from Newton's 'action at a distance' view of physics! Now I understand better what this book means."
And so saying, he opened his book and read, "Don't try to describe motion relative to faraway objects. Physics is simple only when analyzed locally. And locally
the world line that a satellite follows [in spacetime, around the Earth] is already as straight as any world line can be. Forget all this talk about 'deflection' and 'force of gravitation.' I'm inside a spaceship. Or I'm floating outside and near it. Do I feel any 'force of gravitation'? Not at all. Does the spaceship 'feel' such a force? No. Then why talk about it? Recognize that the spaceship and I traverse a region of spacetime free of all force. Acknowledge that the motion through that region is already ideally straight."
The dinner bell was ringing, but still the student sat, musing to himself. "Let me see if I can summarize Einstein's geometric theory of gravity in three ideas: (1) locally, geodesics appear straight; (2) over more extended regions of space and time, geodesics originally receding from each other begin to approach at a rate governed by the curvature of spacetime, and this effect of geometry on matter is what we mean today by that old word 'gravitation'; (3) matter in turn warps geometry. The dimple arises in the apple because the stem is there. I think I see how to put the whole story even more briefly: Space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve. In other words, matter here," he said, rising and picking up the apple by its stem, "curves space here. To produce a curvature in space here is to force a curvature in space there," he went on, as he watched a lingering ant busily following its geodesic a finger's breadth away from the apple's stem. "Thus matter here influences matter there. That is Einstein's explanation for 'gravitation.'"
Then the dinner bell was quiet, and he was gone, with book, magnifying glass-and apple.

§1.2. SPACETIME WITH AND WITHOUT COORDINATES

Now it came to me: . . . the independence of the gravitational acceleration from the nature of the falling substance, may be expressed as follows: In a gravitational field (of small spatial extension) things behave as they do in a space free of gravitation. ... This happened in 1908. Why were another seven years required for the construction of the general theory of relativity? The main reason lies in the fact that it is not so easy to free oneself from the idea that coordinates must have an immediate metrical meaning.
ALBERT EINSTEIN [in Schilpp (1949), pp. 65-67.]
Nothing is more distressing on first contact with the idea of "curved spacetime" than the fear that every simple means of measurement has lost its power in this unfamiliar context. One thinks of oneself as confronted with the task of measuring the shape of a gigantic and fantastically sculptured iceberg as one stands with a meter stick in a tossing rowboat on the surface of a heaving ocean. Were it the rowboat itself whose shape were to be measured, the procedure would be simple enough. One would draw it up on shore, turn it upside down, and drive tacks in lightly at strategic points here and there on the surface. The measurement of distances from tack to
Space tells matter how to move
Matter tells space how to curve
Problem: how to measure in curved spacetime
Figure 1.2.
The crossing of straws in a barn full of hay is a symbol for the world lines that fill up spacetime. By their crossings and bends, these world lines mark events with a uniqueness beyond all need of coordinate systems or coordinates. Typical events symbolized in the diagram, from left to right (black dots), are: absorption of a photon; reemission of a photon; collision between a particle and a particle; collision between a photon and a particle; another collision between a photon and a particle; explosion of a firecracker; and collision of a particle from outside with one of the fragments of that firecracker.
Resolution: characterize events by what happens there
tack would record and reveal the shape of the surface. The precision could be made arbitrarily great by making the number of tacks arbitrarily large. It takes more daring to think of driving several score pitons into the towering iceberg. But with all the daring in the world, how is one to drive a nail into spacetime to mark a point? Happily, nature provides its own way to localize a point in spacetime, as Einstein was the first to emphasize. Characterize the point by what happens there! Give a point in spacetime the name "event." Where the event lies is defined as clearly and sharply as where two straws cross each other in a barn full of hay (Figure 1.2). To say that the event marks a collision of such and such a photon with such and such a particle is identification enough. The world lines of that photon and that particle are rooted in the past and stretch out into the future. They have a rich texture of connections with nearby world lines. These nearby world lines in turn are linked in a hundred ways with world lines more remote. How then does one tell the location of an event? Tell first what world lines participate in the event. Next follow each
Figure 1.3.
Above: Assigning "telephone numbers" to events by way of a system of coordinates. To say that the coordinate system is "smooth" is to say that events which are almost in the same place have almost the same coordinates. Below: Putting the same set of events into equally good order by way of a different system of coordinates. Picked out specially here are two neighboring events: an event named "Q " with coordinates ( x 0 , x 1 ) = ( 77.2 , 22.6 ) x 0 , x 1 = ( 77.2 , 22.6 ) (x^(0),x^(1))=(77.2,22.6)\left(x^{0}, x^{1}\right)=(77.2,22.6)(x0,x1)=(77.2,22.6) and ( x 0 , x 1 ) = ( 18.5 , 51.4 ) x 0 ¯ , x 1 ¯ = ( 18.5 , 51.4 ) (x^( bar(0)),x^( bar(1)))=(18.5,51.4)\left(x^{\overline{0}}, x^{\overline{1}}\right)=(18.5,51.4)(x0,x1)=(18.5,51.4); and an event named " P P P\mathscr{P}P " with coordinates ( x 0 , x 1 ) = ( 79.9 , 20.1 ) x 0 , x 1 = ( 79.9 , 20.1 ) (x^(0),x^(1))=(79.9,20.1)\left(x^{0}, x^{1}\right)=(79.9,20.1)(x0,x1)=(79.9,20.1) and ( x 0 , x 1 ) = ( 18.4 , 47.1 ) x 0 , x 1 = ( 18.4 , 47.1 ) (x^(0),x^(1))=(18.4,47.1)\left(x^{0}, x^{1}\right)=(18.4,47.1)(x0,x1)=(18.4,47.1). Events Q Q Q\mathcal{Q}Q and F F F\mathscr{\mathscr { F }}F are connected by the separation "vector" ξ ξ xi\xiξ. (Precise definition of a vector in a curved spacetime demands going to the mathematical limit in which the two points have an indefinitely small separation [ N N NNN-fold reduction of the separation P Q P Q P-Q\mathscr{P}-\mathscr{Q}PQ ], and, in the resultant locally flat space, multiplying the separation up again by the factor N [ lim N N [ lim N N[lim N rarr ooN[\lim N \rightarrow \inftyN[limN; "tangent space"; "tangent vector"]. Forego here that proper way of stating matters, and forego complete accuracy; hence the quote around the word "vector".) In each coordinate system the separation vector ξ ξ xi\xiξ is characterized by "components" (differences in coordinate values between P P P\mathscr{P}P and Q Q Q\mathscr{Q}Q ):
( ξ 0 , ξ 1 ) = ( 79.9 77.2 , 20.1 22.6 ) = ( 2.7 , 2.5 ) ( ξ 0 , ξ 1 ) = ( 18.4 18.5 , 47.1 51.4 ) = ( 0.1 , 4.3 ) ξ 0 , ξ 1 = ( 79.9 77.2 , 20.1 22.6 ) = ( 2.7 , 2.5 ) ξ 0 , ξ 1 = ( 18.4 18.5 , 47.1 51.4 ) = ( 0.1 , 4.3 ) {:[(xi^(0),xi^(1))=(79.9-77.2","20.1-22.6)=(2.7","-2.5)],[(xi^(0),xi^(1))=(18.4-18.5","47.1-51.4)=(-0.1","-4.3)]:}\begin{aligned} & \left(\xi^{0}, \xi^{1}\right)=(79.9-77.2,20.1-22.6)=(2.7,-2.5) \\ & \left(\xi^{0}, \xi^{1}\right)=(18.4-18.5,47.1-51.4)=(-0.1,-4.3) \end{aligned}(ξ0,ξ1)=(79.977.2,20.122.6)=(2.7,2.5)(ξ0,ξ1)=(18.418.5,47.151.4)=(0.1,4.3)
See Box 1.1 for further discussion of events, coordinates, and vectors.
The name of an event can even be arbitrary
Coordinates provide a convenient naming system
Coordinates generally do not measure length
Several coordinate systems can be used at once
of these world lines. Name the additional events that they encounter. These events pick out further world lines. Eventually the whole barn of hay is catalogued. Each event is named. One can find one's way as surely to a given intersection as the city dweller can pick his path to the meeting of St. James Street and Piccadilly. No numbers. No coordinate system. No coordinates.
That most streets in Japan have no names, and most houses no numbers, illustrates one's ability to do without coordinates. One can abandon the names of two world lines as a means to identify the event where they intersect. Just as one could name a Japanese house after its senior occupant, so one can and often does attach arbitrary names to specific events in spacetime, as in Box 1.1.
Coordinates, however, are convenient. How else from the great thick catalog of events, randomly listed, can one easily discover that along a certain world line one will first encounter event Trinity, then Baker, then Mike, then Argus-but not the same events in some permuted order?
To order events, introduce coordinates! (See Figure 1.3.) Coordinates are four indexed numbers per event in spacetime; on a sheet of paper, only two. Trinity acquires coordinates
( x 0 , x 1 , x 2 , x 3 ) = ( 77 , 23 , 64 , 11 ) . x 0 , x 1 , x 2 , x 3 = ( 77 , 23 , 64 , 11 ) . (x^(0),x^(1),x^(2),x^(3))=(77,23,64,11).\left(x^{0}, x^{1}, x^{2}, x^{3}\right)=(77,23,64,11) .(x0,x1,x2,x3)=(77,23,64,11).
In christening events with coordinates, one demands smoothness but foregoes every thought of mensuration. The four numbers for an event are nothing but an elaborate kind of telephone number. Compare their "telephone" numbers to discover whether two events are neighbors. But do not expect to learn how many meters separate them from the difference in their telephone numbers!
Nothing prevents a subscriber from being served by competing telephone systems, nor an event from being catalogued by alternative coordinate systems (Figure 1.3). Box 1.1 illustrates the relationships between one coordinate system and another, as well as the notation used to denote coordinates and their transformations.
Choose two events, known to be neighbors by the nearness of their coordinate values in a smooth coordinate system. Draw a little arrow from one event to the other. Such an arrow is called a vector. (It is a well-defined concept in flat spacetime, or in curved spacetime in the limit of vanishingly small length; for finite lengths in curved spacetime, it must be refined and made precise, under the new name "tangent vector," on which see Chapter 9.) This vector, like events, can be given a name. But whether named "John" or "Charles" or "Kip," it is a unique, welldefined geometrical object. The name is a convenience, but the vector exists even without it.
Just as a quadruple of coordinates
( x 0 , x 1 , x 2 , x 3 ) = ( 77 , 23 , 64 , 11 ) x 0 , x 1 , x 2 , x 3 = ( 77 , 23 , 64 , 11 ) (x^(0),x^(1),x^(2),x^(3))=(77,23,64,11)\left(x^{0}, x^{1}, x^{2}, x^{3}\right)=(77,23,64,11)(x0,x1,x2,x3)=(77,23,64,11)
is a particularly useful name for the event "Trinity" (it can be used to identify what other events are nearby), so a quadruple of "components"
( ξ 0 , ξ 1 , ξ 2 , ξ 3 ) = ( 1.2 , 0.9 , 0 , 2.1 ) ξ 0 , ξ 1 , ξ 2 , ξ 3 = ( 1.2 , 0.9 , 0 , 2.1 ) (xi^(0),xi^(1),xi^(2),xi^(3))=(1.2,-0.9,0,2.1)\left(\xi^{0}, \xi^{1}, \xi^{2}, \xi^{3}\right)=(1.2,-0.9,0,2.1)(ξ0,ξ1,ξ2,ξ3)=(1.2,0.9,0,2.1)
Box 1.1 MATHEMATICAL NOTATION FOR EVENTS, COORDINATES, AND VECTORS
Events are denoted by capital script, one-letter Latin names such as P , Q , A , B P , Q , A , B P,Q,A,B\mathscr{P}, \mathscr{Q}, \mathscr{A}, \mathscr{B}P,Q,A,B. Sometimes subscripts are used:
P 0 , P 1 , B 6 P 0 , P 1 , B 6 P_(0),P_(1),B_(6)\mathscr{P}_{0}, \mathscr{P}_{1}, \mathscr{B}_{6}P0,P1,B6.
Coordinates of an event P P P\mathscr{P}P are denoted by
t ( P ) , x ( P ) , y ( P ) , z ( P ) t ( P ) , x ( P ) , y ( P ) , z ( P ) t(P),x(P),y(P),z(P)t(\mathscr{P}), x(\mathscr{P}), y(\mathscr{P}), z(\mathscr{P})t(P),x(P),y(P),z(P), or by
or more abstractly by x 0 ( F ) , x 1 ( P ) , x 2 ( P ) x 0 ( F ) , x 1 ( P ) , x 2 ( P ) x^(0)(F),x^(1)(P),x^(2)(P)x^{0}(\mathscr{F}), x^{1}(\mathscr{P}), x^{2}(\mathscr{P})x0(F),x1(P),x2(P), x 3 ( F ) x 3 ( F ) x^(3)(F)x^{3}(\mathscr{F})x3(F),
where it is understood that Greek indices can take on any value 0,1 , 2 , or 3 .
Time coordinate (when one of the four is picked to play this role) x 0 ( P ) x 0 ( P ) x^(0)(P)x^{0}(\mathscr{P})x0(P).
Space coordinates are
and are sometimes denoted by x 1 ( P ) , x 2 ( P ) , x 3 ( P ) x 1 ( P ) , x 2 ( P ) , x 3 ( P ) x^(1)(P),x^(2)(P),x^(3)(P)x^{1}(\mathscr{P}), x^{2}(\mathscr{P}), x^{3}(\mathscr{P})x1(P),x2(P),x3(P)
x j ( P ) x j ( P ) x^(j)(P)x^{j}(\mathscr{P})xj(P) or x k ( P ) x k ( P ) x^(k)(P)x^{k}(\mathscr{P})xk(P) or dots\ldots
Space coordinates are and are sometimes denoted by x^(1)(P),x^(2)(P),x^(3)(P) x^(j)(P) or x^(k)(P) or dots | Space coordinates are | | | :---: | :--- | | and are sometimes denoted by | $x^{1}(\mathscr{P}), x^{2}(\mathscr{P}), x^{3}(\mathscr{P})$ | | $x^{j}(\mathscr{P})$ or $x^{k}(\mathscr{P})$ or $\ldots$ | |
and are sometimes denoted by x j ( P ) x j ( P ) x^(j)(P)x^{j}(\mathscr{P})xj(P) or x k ( P ) x k ( P ) x^(k)(P)x^{k}(\mathscr{P})xk(P) or dots\ldots
It is to be understood that Latin indices take on values 1, 2, or 3 .
Shorthand notation: One soon tires of writing explicitly the functional dependence of the coordinates, x β ( P ) x β ( P ) x^(beta)(P)x^{\beta}(\mathscr{P})xβ(P); so one adopts the shorthand notation for the coordinates of the event P P P\mathscr{P}P, and x β x β x^(beta)x^{\beta}xβ
for the space coordinates. One even begins to think of x β x β x^(beta)x^{\beta}xβ as representing the event P P P\mathscr{P}P itself, but must remind oneself that the values of x 0 , x 1 , x 2 x 0 , x 1 , x 2 x^(0),x^(1),x^(2)x^{0}, x^{1}, x^{2}x0,x1,x2, x 3 x 3 x^(3)x^{3}x3 depend not only on the choice of P P P\mathscr{P}P but also on the arbitrary choice of coordinates!
Other coordinates for the same event P P P\mathscr{P}P may be denoted
x α ¯ ( P ) or just x α ¯ , x α ( P ) or just x α α 2 , x α ^ ( P ) or just x α ^ . x α ¯ ( P )  or just  x α ¯ , x α ( P )  or just  x α α 2 , x α ^ ( P )  or just  x α ^ . {:[x^( bar(alpha))(P)" or just "x^( bar(alpha))","],[x^(alpha^('))(P)" or just "x^(alpha^(alpha_(2)))","],[x^( hat(alpha))(P)" or just "x^( hat(alpha)).]:}\begin{aligned} & x^{\bar{\alpha}}(\mathscr{P}) \text { or just } x^{\bar{\alpha}}, \\ & x^{\alpha^{\prime}}(\mathscr{P}) \text { or just } x^{\alpha^{\alpha_{2}}}, \\ & x^{\hat{\alpha}}(\mathscr{P}) \text { or just } x^{\hat{\alpha}} . \end{aligned}xα¯(P) or just xα¯,xα(P) or just xαα2,xα^(P) or just xα^.
EXAMPLE: In Figure 1.3 ( x 0 , x 1 ) = ( 77.2 , 22.6 ) 1.3 x 0 , x 1 = ( 77.2 , 22.6 ) 1.3(x^(0),x^(1))=(77.2,22.6)1.3\left(x^{0}, x^{1}\right)=(77.2,22.6)1.3(x0,x1)=(77.2,22.6) and ( x 0 , x 1 ) = ( 18.5 , 51.4 ) x 0 ¯ , x 1 ¯ = ( 18.5 , 51.4 ) (x^( bar(0)),x^( bar(1)))=(18.5,51.4)\left(x^{\overline{0}}, x^{\overline{1}}\right)=(18.5,51.4)(x0,x1)=(18.5,51.4) refer to the same event. The bars, primes, and hats distinguish one coordinate system from another; by putting them on the indices rather than on the x x xxx 's, we simplify later notation.
Transformation from one coordinate system to another is achieved by the four functions
which are denoted more succinctly
x 0 ( x 0 , x 1 , x 2 , x 3 ) , x 1 ( x 0 , x 1 , x 2 , x 3 ) , x 2 ( x 0 , x 1 , x 2 , x 3 ) , x 3 ( x 0 , x 1 , x 2 , x 3 ) , x α ¯ ( x β ) x 0 ¯ x 0 , x 1 , x 2 , x 3 , x 1 ¯ x 0 , x 1 , x 2 , x 3 , x 2 ¯ x 0 , x 1 , x 2 , x 3 , x 3 x 0 , x 1 , x 2 , x 3 , x α ¯ x β {:[x^( bar(0))(x^(0),x^(1),x^(2),x^(3))","],[x^( bar(1))(x^(0),x^(1),x^(2),x^(3))","],[x^( bar(2))(x^(0),x^(1),x^(2),x^(3))","],[x^(3)(x^(0),x^(1),x^(2),x^(3))","],[x^( bar(alpha))(x^(beta))]:}\begin{aligned} & x^{\overline{0}}\left(x^{0}, x^{1}, x^{2}, x^{3}\right), \\ & x^{\overline{1}}\left(x^{0}, x^{1}, x^{2}, x^{3}\right), \\ & x^{\overline{2}}\left(x^{0}, x^{1}, x^{2}, x^{3}\right), \\ & x^{3}\left(x^{0}, x^{1}, x^{2}, x^{3}\right), \\ & x^{\bar{\alpha}}\left(x^{\beta}\right) \end{aligned}x0(x0,x1,x2,x3),x1(x0,x1,x2,x3),x2(x0,x1,x2,x3),x3(x0,x1,x2,x3),xα¯(xβ)
Separation vector* (little arrow) reaching from one event Q Q Q\mathscr{Q}Q to neighboring event P P P\mathscr{P}P can be denoted abstractly by
It can also be characterized by the coordinate-value differences \dagger between P P P\mathscr{P}P and Q Q Q\mathscr{Q}Q (called "components" of the vector)
u or v or ξ , or P Q . ξ α x α ( P ) x α ( Q ) , ξ α ¯ x α ¯ ( P ) x α ¯ ( Q ) . u  or  v  or  ξ ,  or  P Q . ξ α x α ( P ) x α ( Q ) , ξ α ¯ x α ¯ ( P ) x α ¯ ( Q ) . {:[u" or "v" or "xi","" or "P-Q.],[xi^(alpha)-=x^(alpha)(P)-x^(alpha)(Q)","],[xi^( bar(alpha))-=x^( bar(alpha)(P)-x^( bar(alpha))(Q).)]:}\begin{aligned} & \boldsymbol{u} \text { or } \boldsymbol{v} \text { or } \xi, \text { or } \mathscr{P}-\mathscr{Q} . \\ & \xi^{\alpha} \equiv x^{\alpha}(\mathscr{P})-x^{\alpha}(\mathscr{Q}), \\ & \xi^{\bar{\alpha}} \equiv x^{\bar{\alpha}(\mathscr{P})-x^{\bar{\alpha}}(\mathscr{Q}) .} \end{aligned}u or v or ξ, or PQ.ξαxα(P)xα(Q),ξα¯xα¯(P)xα¯(Q).
Transformation of components of a vector from one coordinate system to another is achieved by partial derivatives of transformation equations
ξ α ¯ = x α ¯ x β ξ β , ξ α ¯ = x α ¯ x β ξ β , xi^( bar(alpha))=(delx^( bar(alpha)))/(delx^(beta))xi^(beta),\xi^{\bar{\alpha}}=\frac{\partial x^{\bar{\alpha}}}{\partial x^{\beta}} \xi^{\beta},ξα¯=xα¯xβξβ,
since ξ α ¯ = x α ¯ ( P ) x α ¯ ( Q ) = ( x α ¯ / x β ) [ x β ( P ) x β ( Q ) ] ξ α ¯ = x α ¯ ( P ) x α ¯ ( Q ) = x α ¯ / x β x β ( P ) x β ( Q ) xi^( bar(alpha))=x^( bar(alpha))(P)-x^( bar(alpha))(Q)=(delx^( bar(alpha))//delx^(beta))[x^(beta)(P)-x^(beta)(Q)]*†\xi^{\bar{\alpha}}=x^{\bar{\alpha}}(\mathscr{P})-x^{\bar{\alpha}}(\mathscr{Q})=\left(\partial x^{\bar{\alpha}} / \partial x^{\beta}\right)\left[x^{\beta}(\mathscr{P})-x^{\beta}(\mathscr{Q})\right] \cdot \daggerξα¯=xα¯(P)xα¯(Q)=(xα¯/xβ)[xβ(P)xβ(Q)]
Einstein summation convention is used here:
any index that is repeated in a product is automatically summed on
x α ¯ x β ξ β β = 0 3 x α ¯ x β ξ β x α ¯ x β ξ β β = 0 3 x α ¯ x β ξ β (delx^( bar(alpha)))/(delx^(beta))xi^(beta)-=sum_(beta=0)^(3)(delx^( bar(alpha)))/(delx^(beta))xi^(beta)\frac{\partial x^{\bar{\alpha}}}{\partial x^{\beta}} \xi^{\beta} \equiv \sum_{\beta=0}^{3} \frac{\partial x^{\bar{\alpha}}}{\partial x^{\beta}} \xi^{\beta}xα¯xβξββ=03xα¯xβξβ

is a convenient name for the vector "John" that reaches from
( x 0 , x 1 , x 2 , x 3 ) = ( 77 , 23 , 64 , 11 ) x 0 , x 1 , x 2 , x 3 = ( 77 , 23 , 64 , 11 ) (x^(0),x^(1),x^(2),x^(3))=(77,23,64,11)\left(x^{0}, x^{1}, x^{2}, x^{3}\right)=(77,23,64,11)(x0,x1,x2,x3)=(77,23,64,11)
to
( x 0 , x 1 , x 2 , x 3 ) = ( 78.2 , 22.1 , 64.0 , 13.1 ) x 0 , x 1 , x 2 , x 3 = ( 78.2 , 22.1 , 64.0 , 13.1 ) (x^(0),x^(1),x^(2),x^(3))=(78.2,22.1,64.0,13.1)\left(x^{0}, x^{1}, x^{2}, x^{3}\right)=(78.2,22.1,64.0,13.1)(x0,x1,x2,x3)=(78.2,22.1,64.0,13.1)
How to work with the components of a vector is explored in Box 1.1.
There are many ways in which a coordinate system can be imperfect. Figure 1.4 illustrates a coordinate singularity. For another example of a coordinate singularity, run the eye over the surface of a globe to the North Pole. Note the many meridians that meet there ("collapse of cells of egg crates to zero content"). Can't one do better? Find a single coordinate system that will cover the globe without singularity? A theorem says no. Two is the minimum number of "coordinate patches" required to cover the two-sphere without singularity (Figure 1.5). This circumstance emphasizes anew that points and events are primary, whereas coordinates are a mere bookkeeping device.
Figures 1.2 and 1.3 show only a few world lines and events. A more detailed diagram would show a maze of world lines and of light rays and the intersections between them. From such a picture, one can in imagination step to the idealized limit: an infinitely dense collection of light rays and of world lines of infinitesimal test particles. With this idealized physical limit, the mathematical concept of a continuous four-dimensional "manifold" (four-dimensional space with certain smoothness properties) has a one-to-one correspondence; and in this limit continuous, differentiable (i.e., smooth) coordinate systems operate. The mathematics then supplies a tool to reason about the physics.
A simple countdown reveals the dimensionality of the manifold. Take a point P P P\mathscr{P}P in an n n nnn-dimensional manifold. Its neighborhood is an n n nnn-dimensional ball (i.e., the interior of a sphere whose surface has n 1 n 1 n-1n-1n1 dimensions). Choose this ball so that its boundary is a smooth manifold. The dimensionality of this manifold is ( n 1 ) ( n 1 ) (n-1)(n-1)(n1). In this ( n 1 ) ( n 1 ) (n-1)(n-1)(n1)-dimensional manifold, pick a point Q Q Q\mathcal{Q}Q. Its neighborhood is an ( n 1 n 1 n-1n-1n1 )-dimensional ball. Choose this ball so that dots\ldots, and so on. Eventually one comes by this construction to a manifold that is two-dimensional but is not yet known to be two-dimensional (two-sphere). In this two-dimensional manifold, pick a point \Re. Its neighborhood is a two-dimensional ball ("disc"). Choose this disc so that its boundary is a smooth manifold (circle). In this manifold, pick a point R R R\mathscr{R}R. Its neighborhood is a one-dimensional ball, but is not yet known to be one-dimensional ("line segment"). The boundaries of this object are two points. This circumstance tells that the intervening manifold is one-dimensional; therefore the previous manifold was two-dimensional; and so on. The dimensionality of the original manifold is equal to the number of points employed in the construction. For spacetime, the dimensionality is 4 .
This kind of mathematical reasoning about dimensionality makes good sense at the everyday scale of distances, at atomic distances ( 10 8 cm ) 10 8 cm (10^(-8)(cm))\left(10^{-8} \mathrm{~cm}\right)(108 cm), at nuclear dimensions ( 10 13 cm ) 10 13 cm (10^(-13)(cm))\left(10^{-13} \mathrm{~cm}\right)(1013 cm), and even at lengths smaller by several powers of ten, if one judges by the concord between prediction and observation in quantum electrodynamics at high
Figure 1.4.
How a mere coordinate singularity arises. Above: A coordinate system becomes singular when the "cells in the egg crate" are squashed to zero volume. Below: An example showing such a singularity in the Schwarzschild coordinates r , t r , t r,tr, tr,t often used to describe the geometry around a black hole (Chapter 31). For simplicity the angular coordinates θ , ϕ θ , ϕ theta,phi\theta, \phiθ,ϕ have been suppressed. The singularity shows itself in two ways. First, all the points along the dotted line, while quite distinct one from another, are designated by the same pair of ( r , t ) ( r , t ) (r,t)(r, t)(r,t) values; namely, r = 2 m , t = r = 2 m , t = r=2m,t=oor=2 m, t=\inftyr=2m,t=. The coordinates provide no way to distinguish these points. Second, the "cells in the egg crate," of which one is shown grey in the diagram, collapse to zero content at the dotted line. In summary, there is nothing strange about the geometry at the dotted line; all the singularity lies in the coordinate system ("poor system of telephone numbers"). No confusion should be permitted to arise from the accidental circumstance that the t t ttt coordinate attains an infinite value on the dotted line. No such infinity would occur if t t ttt were replaced by the new coordinate t ¯ t ¯ bar(t)\bar{t}t¯, defined by
( t / 2 m ) = tan ( t ¯ / 2 m ) ( t / 2 m ) = tan ( t ¯ / 2 m ) (t//2m)=tan( bar(t)//2m)(t / 2 m)=\tan (\bar{t} / 2 m)(t/2m)=tan(t¯/2m)
When t = t = t=oot=\inftyt=, the new coordinate t ¯ t ¯ bar(t)\bar{t}t¯ is t ¯ = π m t ¯ = π m bar(t)=pi m\bar{t}=\pi mt¯=πm. The r , t ¯ r , t ¯ r, bar(t)r, \bar{t}r,t¯ coordinates still provide no way to distinguish the points along the dotted line. They still give "cells in the egg crate" collapsed to zero content along the dotted line.
Figure 1.5.
Singularities in familiar coordinates on the two-sphere can be eliminated by covering the sphere with two overlapping coordinate patches. A. Spherical polar coordinates, singular at the North and South Poles, and discontinuous at the international date line. B. Projection of the Euclidean coordinates of the Euclidean two-plane, tangent at the North Pole, onto the sphere via a line running to the South Pole; coordinate singularity at the South Pole. C. Coverage of two-sphere by two overlapping coordinate patches. One, constructed as in B, covers without singularity the northern hemisphere and also the southern tropics down to the Tropic of Capricorn. The other (grey) also covers without singularity all of the tropics and the southern hemisphere besides.
energies (corresponding de Broglie wavelength 10 16 cm 10 16 cm 10^(-16)cm10^{-16} \mathrm{~cm}1016 cm ). Moreover, classical general relativity thinks of the spacetime manifold as a deterministic structure, completely well-defined down to arbitrarily small distances. Not so quantum general relativity or "quantum geometrodynamics." It predicts violent fluctuations in the geometry at distances on the order of the Planck length,
L = ( G / c 3 ) 1 / 2 = [ ( 1.054 × 10 27 g cm 2 / sec ) ( 6.670 × 10 8 cm 3 / g sec 2 ) ] 1 / 2 × (1.1) × ( 2.998 × 10 10 cm / sec ) 3 / 2 = 1.616 × 10 33 cm . L = G / c 3 1 / 2 = 1.054 × 10 27 g cm 2 / sec 6.670 × 10 8 cm 3 / g sec 2 1 / 2 × (1.1) × 2.998 × 10 10 cm / sec 3 / 2 = 1.616 × 10 33 cm . {:[L^(**)=(ℏG//c^(3))^(1//2)],[=[(1.054 xx10^(-27)(g)cm^(2)//sec)(6.670 xx10^(-8)cm^(3)//gsec^(2))]^(1//2)xx],[(1.1) xx(2.998 xx10^(10)(cm)//sec)^(-3//2)],[=1.616 xx10^(-33)cm.]:}\begin{align*} & L^{*}=\left(\hbar G / c^{3}\right)^{1 / 2} \\ & =\left[\left(1.054 \times 10^{-27} \mathrm{~g} \mathrm{~cm}^{2} / \mathrm{sec}\right)\left(6.670 \times 10^{-8} \mathrm{~cm}^{3} / \mathrm{g} \mathrm{sec}^{2}\right)\right]^{1 / 2} \times \\ & \times\left(2.998 \times 10^{10} \mathrm{~cm} / \mathrm{sec}\right)^{-3 / 2} \tag{1.1}\\ & =1.616 \times 10^{-33} \mathrm{~cm} . \end{align*}L=(G/c3)1/2=[(1.054×1027 g cm2/sec)(6.670×108 cm3/gsec2)]1/2×(1.1)×(2.998×1010 cm/sec)3/2=1.616×1033 cm.
No one has found any way to escape this prediction. As nearly as one can estimate, these fluctuations give space at small distances a "multiply connected" or "foamlike" character. This lack of smoothness may well deprive even the concept of dimensionality itself of any meaning at the Planck scale of distances. The further exploration of this issue takes one to the frontiers of Einstein's theory (Chapter 44).
If spacetime at small distances is far from the mathematical model of a continuous manifold, is there not also at larger distances a wide gap between the mathematical
idealization and the physical reality? The infinitely dense collection of light rays and of world lines of infinitesimal test particles that are to define all the points of the manifold: they surely are beyond practical realization. Nobody has ever found a particle that moves on timelike world lines (finite rest mass) lighter than an electron. A collection of electrons, even if endowed with zero density of charge ( e + e + e^(+)\mathrm{e}^{+}e+and e e e^(-)\mathrm{e}^{-}e world lines present in equal numbers) will have a density of mass. This density will curve the very manifold under study. Investigation in infinite detail means unlimited density, and unlimited disturbance of the geometry.
However, to demand investigatability in infinite detail in the sense just described is as out of place in general relativity as it would be in electrodynamics or gas dynamics. Electrodynamics speaks of the strength of the electric and magnetic field at each point in space and at each moment of time. To measure those fields, it is willing to contemplate infinitesimal test particles scattered everywhere as densely as one pleases. However, the test particles do not have to be there at all to give the field reality. The field has everywhere a clear-cut value and goes about its deterministic dynamic evolution willy-nilly and continuously, infinitesimal test particles or no infinitesimal test particles. Similarly with the geometry of space.
In conclusion, when one deals with spacetime in the context of classical physics, one accepts (1) the notion of "infinitesimal test particle" and (2) the idealization that the totality of identifiable events forms a four-dimensional continuous manifold. Only at the end of this book will a look be taken at some of the limitations placed by the quantum principle on one's way of speaking about and analyzing spacetime.

§1.3. WEIGHTLESSNESS

"Gravity is a great mystery. Drop a stone. See it fall. Hear it hit. No one understands why." What a misleading statement! Mystery about fall? What else should the stone do except fall? To fall is normal. The abnormality is an object standing in the way of the stone. If one wishes to pursue a "mystery," do not follow the track of the falling stone. Look instead at the impact, and ask what was the force that pushed the stone away from its natural "world line," (i.e., its natural track through spacetime). That could lead to an interesting issue of solid-state physics, but that is not the topic of concern here. Fall is. Free fall is synonymous with weightlessness: absence of any force to drive the object away from its normal track through spacetime. Travel aboard a freely falling elevator to experience weightlessness. Or travel aboard a spaceship also falling straight toward the Earth. Or, more happily, travel aboard a spaceship in that state of steady fall toward the Earth that marks a circular orbit. In each case one is following a natural track through spacetime.
The traveler has one chemical composition, the spaceship another; yet they travel together, the traveler weightless in his moving home. Objects of such different nuclear constitution as aluminum and gold fall with accelerations that agree to better than one part in 10 11 10 11 10^(11)10^{11}1011, according to Roll, Krotkov, and Dicke (1964), one of the most important null experiments in all physics (see Figure 1.6). Individual molecules fall in step, too, with macroscopic objects [Estermann, Simpson, and Stern (1938)]; and so do individual neutrons [Dabbs, Harvey, Paya, and Horstmann (1965)], individual
Difficulty in defining geometry even at classical distances?
No; one must accept geometry at classical distances as meaningful
Free fall is the natural state of motion
All objects fall with the same acceleration
Figure 1.6.
Principle of the Roll-Krotkov-Dicke experiment, which showed that the gravitational accelerations of gold and aluminum are equal to 1 part in 10 11 10 11 10^(11)10^{11}1011 or better (Princeton, 1964). In the upper lefthand corner, equal masses of gold and aluminum hang from a supporting bar. This bar in turn is supported at its midpoint. If both objects fall toward the sun with the same acceleration of g = 0.59 cm / sec 2 g = 0.59 cm / sec 2 g=0.59cm//sec^(2)g=0.59 \mathrm{~cm} / \mathrm{sec}^{2}g=0.59 cm/sec2, the bar does not turn. If the Au mass receives a higher acceleration, g + δ g g + δ g g+delta gg+\delta gg+δg, then the gold end of the bar starts to turn toward the sun in the Earth-fixed frame. Twelve hours later the sun is on the other side, pulling the other way. The alternating torque lends itself to recognition against a background of noise because of its precise 24 -hour period. Unhappily, any substantial mass nearby, such as an experimenter, located at M M MMM, will produce a torque that swamps the effect sought. Therefore the actual arrangement was as shown in the body of the figure. One gold weight and two aluminum weights were supported at the three corners of a horizontal equilateral triangle, 6 cm on a side (three-fold axis of symmetry, giving zero response to all the simplest nonuniformities in the gravitational field). Also, the observers performed all operations remotely to eliminate their own gravitational effects*. To detect a rotation of the torsion balance as small as 10 9 10 9 ∼10^(-9)\sim 10^{-9}109 rad without disturbing the balance, Roll, Krotkov, and Dicke reflected a very weak light beam from the optically flat back face of the quartz triangle. The image of the source slit fell on a wire of about the same size as the slit image. The light transmitted past the wire fell on a photomultiplier. A separate oscillator circuit drove the wire back and forth across the image at 3,000 hertz. When the image was centered perfectly, only even harmonics of the oscillation frequency appeared in the light intensity. However, when the image was displaced slightly to one side, the fundamental frequency appeared in the light intensity. The electrical output of the photomultiplier then contained a 3,000 -hertz component. The magnitude and sign of this component were determined automatically. Equally automatically a proportional D.C. voltage was applied to the electrodes shown in the diagram. It restored the torsion balance to its zero position. The D.C. voltage required to restore the balance to its zero position was recorded as a measure of the torque acting on the pendulum. This torque was Fourier-analyzed over a period of many days. The magnitude of the Fourier component of 24 -hour period indicated a ratio δ g / g = ( 0.96 ± 1.04 ) × 10 11 δ g / g = ( 0.96 ± 1.04 ) × 10 11 delta g//g=(0.96+-1.04)xx10^(-11)\delta g / g=(0.96 \pm 1.04) \times 10^{-11}δg/g=(0.96±1.04)×1011. Aluminum and gold thus fall with the same acceleration, despite their important differences summarized in the table.
Ratios A l A l AlA lAl A u A u AuA uAu
Number of neutrons Number of protons  Number of neutrons   Number of protons  (" Number of neutrons ")/(" Number of protons ")\frac{\text { Number of neutrons }}{\text { Number of protons }} Number of neutrons  Number of protons  1.08 1.5
Mass of kinetic energy of K-electron Rest mass of electron  Mass of kinetic energy of K-electron   Rest mass of electron  (" Mass of kinetic energy of K-electron ")/(" Rest mass of electron ")\frac{\text { Mass of kinetic energy of K-electron }}{\text { Rest mass of electron }} Mass of kinetic energy of K-electron  Rest mass of electron  0.005 0.16
Electrostatic mass-energy of nucleus Mass of atom  Electrostatic mass-energy of nucleus   Mass of atom  (" Electrostatic mass-energy of nucleus ")/(" Mass of atom ")\frac{\text { Electrostatic mass-energy of nucleus }}{\text { Mass of atom }} Electrostatic mass-energy of nucleus  Mass of atom  0.001 0.004
Ratios Al Au (" Number of neutrons ")/(" Number of protons ") 1.08 1.5 (" Mass of kinetic energy of K-electron ")/(" Rest mass of electron ") 0.005 0.16 (" Electrostatic mass-energy of nucleus ")/(" Mass of atom ") 0.001 0.004| Ratios | $A l$ | $A u$ | | :--- | :---: | :---: | | $\frac{\text { Number of neutrons }}{\text { Number of protons }}$ | 1.08 | 1.5 | | $\frac{\text { Mass of kinetic energy of K-electron }}{\text { Rest mass of electron }}$ | 0.005 | 0.16 | | $\frac{\text { Electrostatic mass-energy of nucleus }}{\text { Mass of atom }}$ | 0.001 | 0.004 |
The theoretical implications of this experiment will be discussed in greater detail in Chapters 16 and 38.
Braginsky and Panov (1971) at Moscow University performed an experiment identical in principle to that of Dicke-Roll-Krotkov, but with a modified experimental set-up. Comparing the accelerations of platinum and aluminum rather than of gold and aluminum, they say that
δ g / g 1 × 10 12 δ g / g 1 × 10 12 delta g//g <= 1xx10^(-12)\delta g / g \leq 1 \times 10^{-12}δg/g1×1012
*Other perturbations had to be, and were, guarded against. (1) A bit of iron on the torsion balance as big as 10 3 cm 10 3 cm 10^(-3)cm10^{-3} \mathrm{~cm}103 cm on a side would have contributed, in the Earth's magnetic field, a torque a hundred times greater than the measured torque. (2) The unequal pressure of radiation on the two sides of a mass would have produced an unacceptably large perturbation if the temperature difference between these two sides had exceeded 10 4 K 10 4 K 10^(-4)^(@)K10^{-4}{ }^{\circ} \mathrm{K}104K. (3) Gas evolution from one side of a mass would have propelled it like a rocket. If the rate of evolution were as great as 10 8 g / day 10 8 g / day 10^(-8)g//day10^{-8} \mathrm{~g} / \mathrm{day}108 g/day, the calculated force would have been 10 7 g cm / sec 2 10 7 g cm / sec 2 ∼10^(-7)gcm//sec^(2)\sim 10^{-7} \mathrm{~g} \mathrm{~cm} / \mathrm{sec}^{2}107 g cm/sec2, enough to affect the measurements. (4) The rotation was measured with respect to the pier that supported the equipment. As a guarantee that this pier did not itself rotate, it was anchored to bed rock. (5) Electrostatic forces were eliminated; otherwise they would have perturbed the balance.

electrons [Witteborn and Fairbank (1967)] and individual mu mesons [Beall (1970)]. What is more, not one of these objects has to see out into space to know how to move.
Contemplate the interior of a spaceship, and a key, penny, nut, and pea by accident or design set free inside. Shielded from all view of the world outside by the walls of the vessel, each object stays at rest relative to the vessel. Or it moves through the room in a straight line with uniform velocity. That is the lesson which experience shouts out.
Forego talk of acceleration! That, paradoxically, is the lesson of the circumstance that "all objects fall with the same acceleration." Whose fault were those accelerations, after all? They came from allowing a groundbased observer into the act. The

Box 1.2 MATERIALS OF THE MOST DIVERSE COMPOSITION FALL WITH THE SAME ACCELERATION ('STANDARD WORLD LINE")

Aristotle: "the downward movement of a mass of gold or lead, or of any other body endowed with weight, is quicker in proportion to its size."
Pre-Galilean literature: metal and wood weights fall at the same rate.
Galileo: (1) "the variation of speed in air between balls of gold, lead, copper, porphyry, and other heavy materials is so slight that in a fall of 100 cubits [about 46 meters] a ball of gold would surely not outstrip one of copper by as much as four fingers. Having observed this, I came to the conclusion that in a medium totally void of resistance all bodies would fall with the same speed." (2) later experiments of greater precision "diluting gravity" and finding same time of descent for different objects along an inclined plane.
Newton: inclined plane replaced by arc of pendulum bob; "time of fall" for bodies of different composition determined by comparing time of oscillation of pendulum bobs of the two materials. Ultimate limit of precision in such experiments limited by problem of determining effective length of each pendulum: ( ( quad(\quad(( acceleration ) = ( 2 π / ) = ( 2 π / )=(2pi//)=(2 \pi /)=(2π/ period) 2 2 ^(2){ }^{2}2 (length).
Lorand von Eötvös, Budapest, 1889 and 1922: compared on the rotating earth the vertical defined by a plumb bob of one material with the vertical defined by a plumb bob of other material. The two hanging masses, by the two unbroken threads that support them, were drawn along identical world lines through spacetime (middle of the laboratory of Eötvös!). If cut free, would they also follow identical tracks through spacetime ("normal world line of test mass")? If so, the acceleration that draws the actual world line from the normal free-fall world line will have a standard value, a a a\boldsymbol{a}a. The experiment of Eötvös did not try to test agreement on the magnitude of a a a\boldsymbol{a}a between the two masses. Doing so would have required (1) cutting the threads and (2) following the fall of the two masses. Eötvös renounced this approach in favor of a static observation that he could make with greater precision, comparing the direction of a a a\boldsymbol{a}a for the two masses. The direction of the supporting thread, so his argument ran, reveals the direction in which the mass is being dragged away from its normal world line of "free fall" or "weightlessness." This acceleration is the vectorial resultant of (1) an acceleration of magnitude g g g\boldsymbol{g}g, directed outward against so-called gravity, and (2) an acceleration directed toward the axis of rotation of the earth, of magnitude ω 2 R sin θ ω 2 R sin θ omega^(2)R sin theta\omega^{2} R \sin \thetaω2Rsinθ ( ω ω omega\omegaω, angular ve-
push of the ground under his feet was driving him away from a natural world line. Through that flaw in his arrangements, he became responsible for all those accelerations. Put him in space and strap rockets to his legs. No difference!* Again the responsibility for what he sees is his. Once more he notes that "all objects fall with
locity; R R RRR, radius of earth; θ θ theta\thetaθ, polar angle measured from North Pole to location of experiment). This centripetal acceleration has a vertical component ω 2 R sin 2 θ ω 2 R sin 2 θ -omega^(2)Rsin^(2)theta-\omega^{2} R \sin ^{2} \thetaω2Rsin2θ too small to come into discussion. The important component is ω 2 R sin θ cos θ ω 2 R sin θ cos θ omega^(2)R sin theta cos theta\omega^{2} R \sin \theta \cos \thetaω2Rsinθcosθ, directed northward and parallel to the surface of the earth. It deflects the thread by the angle
horizontal acceleration
vertical acceleration
$$
= ω 2 R sin θ cos θ g = 3.4 cm / sec 2 980 cm / sec 2 sin θ cos θ = 1.7 × 10 3 radian at θ = 45 = ω 2 R sin θ cos θ g = 3.4 cm / sec 2 980 cm / sec 2 sin θ cos θ = 1.7 × 10 3  radian at  θ = 45 {:[=(omega^(2)R sin theta cos theta)/(g)],[=(3.4(cm)//sec^(2))/(980(cm)//sec^(2))sin theta cos theta],[=1.7 xx10^(-3)" radian at "theta=45^(@)]:}\begin{aligned} & =\frac{\omega^{2} R \sin \theta \cos \theta}{g} \\ & =\frac{3.4 \mathrm{~cm} / \mathrm{sec}^{2}}{980 \mathrm{~cm} / \sec ^{2}} \sin \theta \cos \theta \\ & =1.7 \times 10^{-3} \text { radian at } \theta=45^{\circ} \end{aligned}=ω2Rsinθcosθg=3.4 cm/sec2980 cm/sec2sinθcosθ=1.7×103 radian at θ=45
$$
from the straight line connecting the center of the earth to the point of support. A difference, δ g δ g delta g\delta gδg, of one part in 10 8 10 8 10^(8)10^{8}108 between g g ggg for the two hanging substances would produce a difference in angle of hang of plumb bobs equal to 1.7 × 10 11 1.7 × 10 11 1.7 xx10^(-11)1.7 \times 10^{-11}1.7×1011 radian at Budapest ( θ = 42.5 ) θ = 42.5 (theta=42.5^(@))\left(\theta=42.5^{\circ}\right)(θ=42.5). Eötvös reported δ g / g δ g / g delta g//g\delta g / gδg/g less than a few parts in 10 9 10 9 10^(9)10^{9}109.
Roll, Krotkov, and Dicke, Princeton, 1964: employed as fiducial acceleration, not the 1.7 cm / sec 2 1.7 cm / sec 2 1.7cm//sec^(2)1.7 \mathrm{~cm} / \mathrm{sec}^{2}1.7 cm/sec2 steady horizontal acceleration, produced by the earth's rotation at θ = 45 θ = 45 theta=45^(@)\theta=45^{\circ}θ=45, but the daily alternat-
ing 0.59 cm / sec 2 0.59 cm / sec 2 0.59cm//sec^(2)0.59 \mathrm{~cm} / \mathrm{sec}^{2}0.59 cm/sec2 produced by the sun's attraction. Reported | g ( Au ) g ( Al ) | / g | g ( Au ) g ( Al ) | / g |g(Au)-g(Al)|//g|g(\mathrm{Au})-g(\mathrm{Al})| / g|g(Au)g(Al)|/g less than 1 × 10 11 1 × 10 11 1xx10^(-11)1 \times 10^{-11}1×1011. See Figure 1.6.
Braginsky and Panov, Moscow, 1971: like Roll, Krotkov, and Dicke, employed Sun's attraction as fiducial acceleration. Reported | g ( Pt ) g ( Al ) | / g | g ( Pt ) g ( Al ) | / g |g(Pt)-g(Al)|//g|g(\mathrm{Pt})-g(\mathrm{Al})| / g|g(Pt)g(Al)|/g less than 1 × 10 12 1 × 10 12 1xx10^(-12)1 \times 10^{-12}1×1012.
Beall, 1970: particles that are deflected less by the Earth's or the sun's gravitational field than a photon would be, effectively travel faster than light. If they are charged or have other electromagnetic structure, they would then emit Čerenkov radiation, and reduce their velocity below threshold in less than a micron of travel. The threshold is at energies around 10 3 mc 2 10 3 mc 2 10^(3)mc^(2)10^{3} \mathrm{mc}^{2}103mc2. Ultrarelativistic particles in cosmic-ray showers are not easily identified, but observations of 10 13 eV 10 13 eV 10^(13)eV10^{13} \mathrm{eV}1013eV muons show that muons are not "too light" by as much as 5 × 10 5 5 × 10 5 5xx10^(-5)5 \times 10^{-5}5×105. Conversely, a particle P P PPP bound more strongly than photons by gravity will transfer the momentum needed to make pair production γ P + P γ P + P ¯ gamma rarrP+ bar(P)\gamma \rightarrow \mathrm{P}+\overline{\mathrm{P}}γP+P occur within a submicron decay length. The existence of photons with energies above 10 13 eV 10 13 eV 10^(13)eV10^{13} \mathrm{eV}1013eV shows that e ± e ± e^(+-)\mathrm{e}^{ \pm}e±are not "too heavy" by 5 parts in 10 9 , μ ± 10 9 , μ ± 10^(9),mu^(+-)10^{9}, \mu^{ \pm}109,μ±not by 2 in 10 4 , Λ , Ξ , Ω 10 4 , Λ , Ξ , Ω 10^(4),Lambda,Xi^(-),Omega^(-)10^{4}, \Lambda, \Xi^{-}, \Omega^{-}104,Λ,Ξ,Ωnot by a few per cent.
Figure 1.7.
"Weightlessness" as test for a local inertial frame of reference ("Lorentz frame"). Each spring-driven cannon succeeds in driving its projectile, a steel ball bearing, through the aligned holes in the sheets of lucite, and into the woven-mesh pocket, when the frame of reference is free of rotation and in free fall ("normal world line through spacetime"). A cannon would fail (curved and ricocheting trajectory at bottom of drawing) if the frame were hanging as indicated when the cannon went off ("frame drawn away by pull of rope from its normal world line through spacetime"). Harold Waage at Princeton has constructed such a model for an inertial reference frame with lucite sheets about 1 m square. The "fuses" symbolizing time delay were replaced by electric relays. Penetration fails if the frame (1) rotates, (2) accelerates, or (3) does any combination of the two. It is difficult to cite any easily realizable device that more fully illustrates the meaning of the term "local Lorentz frame."
the same acceleration." Physics looks as complicated to the jet-driven observer as it does to the man on the ground. Rule out both observers to make physics look simple. Instead, travel aboard the freely moving spaceship. Nothing could be more natural than what one sees: every free object moves in a straight line with uniform velocity. This is the way to do physics! Work in a very special coordinate system:
Eliminate the acceleration by use of a local inertial frame a coordinate frame in which one is weightless; a local inertial frame of reference. Or calculate how things look in such a frame. Or-if one is constrained to a groundbased frame of reference - use a particle moving so fast, and a path length so limited, that the ideal, freely falling frame of reference and the actual ground-based frame get out of alignment by an amount negligible on the scale of the experiment. [Given a 1 , 500 m 1 , 500 m 1,500-m1,500-\mathrm{m}1,500m linear accelerator, and a 1 GeV electron, time of flight ( 1.5 × 10 5 cm ) 1.5 × 10 5 cm ≃(1.5 xx10^(5)(cm))\simeq\left(1.5 \times 10^{5} \mathrm{~cm}\right)(1.5×105 cm) /
( 3 × 10 10 cm / sec ) = 0.5 × 10 5 sec 3 × 10 10 cm / sec = 0.5 × 10 5 sec (3xx10^(10)(cm)//sec)=0.5 xx10^(-5)sec\left(3 \times 10^{10} \mathrm{~cm} / \mathrm{sec}\right)=0.5 \times 10^{-5} \mathrm{sec}(3×1010 cm/sec)=0.5×105sec; fall in this time 1 2 g t 2 = ( 490 cm / sec 2 ) ( 0.5 × 1 2 g t 2 = 490 cm / sec 2 ( 0.5 × ∼(1)/(2)gt^(2)=(490(cm)//sec^(2))(0.5 xx\sim \frac{1}{2} g t^{2}=\left(490 \mathrm{~cm} / \mathrm{sec}^{2}\right)(0.5 \times12gt2=(490 cm/sec2)(0.5× 10 5 sec ) 2 10 8 cm 10 5 sec 2 10 8 cm 10^(-5)sec)^(2)≃10^(-8)cm\left.10^{-5} \mathrm{sec}\right)^{2} \simeq 10^{-8} \mathrm{~cm}105sec)2108 cm.]
In analyzing physics in a local inertial frame of reference, or following an ant on his little section of apple skin, one wins simplicity by foregoing every reference to what is far away. Physics is simple only when viewed locally: that is Einstein's great lesson.
Newton spoke differently: "Absolute space, in its own nature, without relation to anything external, remains always similar and immovable." But how does one give meaning to Newton's absolute space, find its cornerstones, mark out its straight lines? In the real world of gravitation, no particle ever follows one of Newton's straight lines. His ideal geometry is beyond observation. "A comet going past the sun is deviated from an ideal straight line." No. There is no pavement on which to mark out that line. The "ideal straight line" is a myth. It never happened, and it never will.
"It required a severe struggle [for Newton] to arrive at the concept of independent and absolute space, indispensible for the development of theory. . . Newton's decision was, in the contemporary state of science, the only possible one, and particularly the only fruitful one. But the subsequent development of the problems, proceeding in a roundabout way which no one could then possibly foresee, has shown that the resistance of Leibniz and Huygens, intuitively well-founded but supported by inadequate arguments, was actually justified. ... It has required no less strenuous exertions subsequently to overcome this concept [of absolute space]"
[A. EINSTEIN (1954)].
What is direct and simple and meaningful, according to Einstein, is the geometry in every local inertial reference frame. There every particle moves in a straight line with uniform velocity. Define the local inertial frame so that this simplicity occurs for the first few particles (Figure 1.7). In the frame thus defined, every other free particle is observed also to move in a straight line with uniform velocity. Collision and disintegration processes follow the laws of conservation of momentum and energy of special relativity. That all these miracles come about, as attested by tens of thousands of observations in elementary particle physics, is witness to the inner workings of the machinery of the world. The message is easy to summarize: (1) physics is always and everywhere locally Lorentzian; i.e., locally the laws of special relativity are valid; (2) this simplicity shows most clearly in a local Lorentz frame of reference ("inertial frame of reference"; Figure 1.7); and (3) to test for a local Lorentz frame, test for weightlessness!

§1.4. LOCAL LORENTZ GEOMETRY, WITH AND WITHOUT COORDINATES

On the surface of an apple within the space of a thumbprint, the geometry is Euclidean (Figure 1.1; the view in the magnifying glass). In spacetime, within a limited region, the geometry is Lorentzian. On the apple the distances between point and point accord with the theorems of Euclid. In spacetime the intervals ("proper distance," "proper time") between event and event satisfy the corresponding theorems of Lorentz-Minkowski geometry (Box 1.3). These theorems lend themselves
Newton's absolute space is unobservable, nonexistent
But Einstein's local inertial frames exist, are simple
In local inertial frames, physics is Lorentzian
Local Lorentz geometry is the spacetime analog of local Euclidean geometry.
Box 1.3 LOCAL LORENTZ GEOMETRY AND LOCAL EUCLIDEAN GEOMETRY: WITH AND WITHOUT COORDINATES

I. Local Euclidean Geometry

What does it mean to say that the geometry of a tiny thumbprint on the apple is Euclidean?
A. Coordinate-free language (Euclid):
Given a line P C P C PC\mathscr{P C}PC. Extend it by an equal distance C Z C Z CZ\mathcal{C} \mathscr{Z}CZ. Let B B B\mathscr{B}B be a point not on A Z A Z AZ\mathscr{A} \mathscr{Z}AZ but equidistant from a a aaa and \mathscr{\sim}. Then
s a S 2 = s a e 2 + s s s e 2 s a S 2 = s a e 2 + s s s e 2 s_(aS)^(2)=s_(ae^(2))+s_(sse)^(2)s_{a \mathscr{S}}{ }^{2}=s_{a e^{2}}+s_{s s e}{ }^{2}saS2=sae2+ssse2

(Theorem of Pythagoras; also other theorems of Euclidean geometry.)
B. Language of coordinates (Descartes):
From any point a a aaa to any other point B B B\mathscr{B}B there is a distance s s sss given in suitable (Euclidean) coordinates by
s Q S 2 = [ x 1 ( B ) x 1 ( C ) ] 2 + [ x 2 ( B ) x 2 ( C ) ] 2 s Q S 2 = x 1 ( B ) x 1 ( C ) 2 + x 2 ( B ) x 2 ( C ) 2 s_(QS)^(2)=[x^(1)(B)-x^(1)(C)]^(2)+[x^(2)(B)-x^(2)(C)]^(2)s_{\mathscr{Q} \mathscr{S}}{ }^{2}=\left[x^{1}(\mathscr{B})-x^{1}(\mathscr{C})\right]^{2}+\left[x^{2}(\mathscr{B})-x^{2}(\mathscr{C})\right]^{2}sQS2=[x1(B)x1(C)]2+[x2(B)x2(C)]2.
If one succeeds in finding any coordinate system where this is true for all points a a aaa and B B B\mathscr{B}B in the thumbprint, then one is guaranteed that (i) this coordinate system is locally Euclidean, and (ii) the geometry of the apple's surface is locally Euclidean.

II. Local Lorentz Geometry

What does it mean to say that the geometry of a sufficiently limited region of spacetime in the real physical world is Lorentzian?
A. Coordinate-free language (Robb 1936):
Let a Z a Z aZa \mathscr{Z}aZ be the world line of a free particle. Let B B B\mathscr{B}B be an event not on this world line. Let a light ray from B B B\mathscr{B}B strike A A A\mathscr{A}A at the event Q Q Q\mathcal{Q}Q. Let a light ray take off from such an earlier event P P P\mathscr{P}P along A Z A Z AZ\mathscr{A} \mathscr{Z}AZ that it reaches B B B\mathscr{B}B. Then the proper distance s G 5 s G 5 s_(G5)s_{\mathscr{G} 5}sG5 (spacelike separation) or proper time τ d : g τ d : g tau_(d:g)\tau_{d: g}τd:g (timelike separation) is given by
s d 2 τ d b 2 = τ d Q τ d φ s d 2 τ d b 2 = τ d Q τ d φ s_(dक勹)^(2)-=-tau_(dकb)^(2)=-tau_(dQ)tau_(d varphi)s_{d क 勹}^{2} \equiv-\tau_{d क b}^{2}=-\tau_{d Q} \tau_{d \varphi}sd2τdb2=τdQτdφ
Proof of above criterion for local Lorentz geometry, using coordinate methods in the local Lorentz frame where particle remains at rest:
τ d s 2 = t 2 x 2 = ( t x ) ( t + x ) = τ d φ τ d Q . τ d s 2 = t 2 x 2 = ( t x ) ( t + x ) = τ d φ τ d Q . {:[tau_(dकs)^(2)=t^(2)-x^(2)=(t-x)(t+x)],[=tau_(d varphi)tau_(dQ).]:}\begin{aligned} \tau_{d क s}{ }^{2} & =t^{2}-x^{2}=(t-x)(t+x) \\ & =\tau_{d \varphi} \tau_{d Q} . \end{aligned}τds2=t2x2=(tx)(t+x)=τdφτdQ.
B. Language of coordinates (Lorentz, Poincaré, Minkowski, Einstein):
From any event a a aaa to any other nearby event B B B\mathscr{B}B, there is a proper distance s C σ s C σ s_(C in sigma)s_{C \in \sigma}sCσ or proper time τ G : 98 τ G : 98 tau_(G:98)\tau_{G: 98}τG:98 given in suitable (local Lorentz) coordinates by
s G B 2 = τ G S 2 = [ x 0 ( G ) x 0 ( C ) ] 2 + [ x 1 ( B ) x 1 ( Q ) ] 2 + [ x 2 ( B ) x 2 ( Q ) ] 2 + [ x 3 ( B ) x 3 ( Q ) ] 2 . s G B 2 = τ G S 2 = x 0 ( G ) x 0 ( C ) 2 + x 1 ( B ) x 1 ( Q ) 2 + x 2 ( B ) x 2 ( Q ) 2 + x 3 ( B ) x 3 ( Q ) 2 . {:[s_(GB)^(2)=-tau_(GS)^(2)=-[x^(0)(G)-x^(0)(C)]^(2)],[+[x^(1)(B)-x^(1)(Q)]^(2)],[+[x^(2)(B)-x^(2)(Q)]^(2)],[+[x^(3)(B)-x^(3)(Q)]^(2).]:}\begin{aligned} s_{G \mathscr{B}}^{2}=-\tau_{\mathscr{G S}}^{2}= & -\left[x^{0}(\mathscr{G})-x^{0}(\mathscr{C})\right]^{2} \\ & +\left[x^{1}(\mathscr{B})-x^{1}(\mathscr{Q})\right]^{2} \\ & +\left[x^{2}(\mathscr{B})-x^{2}(\mathscr{Q})\right]^{2} \\ & +\left[x^{3}(\mathscr{B})-x^{3}(\mathscr{Q})\right]^{2} . \end{aligned}sGB2=τGS2=[x0(G)x0(C)]2+[x1(B)x1(Q)]2+[x2(B)x2(Q)]2+[x3(B)x3(Q)]2.
If one succeeds in finding any coordinate system where this is locally true for all neighboring events a a aaa and B B B\mathscr{B}B, then one is guaranteed that (i) this coordinate system is locally Lorentzian, and (ii) the geometry of spacetime is locally Lorentzian.

III. Statements of Fact

The geometry of an apple's surface is locally Euclidean everywhere. The geometry of spacetime is locally Lorentzian everywhere.
Box 1.3 (continued)

IV. Local Geometry in the Language of Modern Mathematics

A. The metric for any manifold:
At each point on the apple, at each event of spacetime, indeed, at each point of any "Riemannian manifold," there exists a geometrical object called the metric tensor g g g\boldsymbol{g}g. It is a machine with two input slots for the insertion of two vectors:
If one inserts the same vector u u u\boldsymbol{u}u into both slots, one gets out the square of the length of u u u\boldsymbol{u}u :
g ( u , u ) = u 2 g ( u , u ) = u 2 g(u,u)=u^(2)\boldsymbol{g}(\boldsymbol{u}, \boldsymbol{u})=\boldsymbol{u}^{2}g(u,u)=u2
If one inserts two different vectors, u u u\boldsymbol{u}u and v v v\boldsymbol{v}v (it matters not in which order!), one gets out a number called the "scalar product of u u u\boldsymbol{u}u on v v v\boldsymbol{v}v " and denoted u v u v u*v\boldsymbol{u} \cdot \boldsymbol{v}uv :
g ( u , v ) = g ( v , u ) = u v = v u g ( u , v ) = g ( v , u ) = u v = v u g(u,v)=g(v,u)=u*v=v*u\boldsymbol{g}(\boldsymbol{u}, \boldsymbol{v})=\boldsymbol{g}(\boldsymbol{v}, \boldsymbol{u})=\boldsymbol{u} \cdot \boldsymbol{v}=\boldsymbol{v} \cdot \boldsymbol{u}g(u,v)=g(v,u)=uv=vu
The metric is a linear machine:
g ( 2 u + 3 w , v ) = 2 g ( u , v ) + 3 g ( w , v ) g ( u , a v + b w ) = a g ( u , v ) + b g ( u , w ) g ( 2 u + 3 w , v ) = 2 g ( u , v ) + 3 g ( w , v ) g ( u , a v + b w ) = a g ( u , v ) + b g ( u , w ) {:[g(2u+3w","v)=2g(u","v)+3g(w","v)],[g(u","av+bw)=ag(u","v)+bg(u","w)]:}\begin{aligned} & \boldsymbol{g}(2 \boldsymbol{u}+3 \boldsymbol{w}, \boldsymbol{v})=2 \boldsymbol{g}(\boldsymbol{u}, \boldsymbol{v})+3 \boldsymbol{g}(\boldsymbol{w}, \boldsymbol{v}) \\ & \boldsymbol{g}(\boldsymbol{u}, a \boldsymbol{v}+b \boldsymbol{w})=a \boldsymbol{g}(\boldsymbol{u}, \boldsymbol{v})+b \boldsymbol{g}(\boldsymbol{u}, \boldsymbol{w}) \end{aligned}g(2u+3w,v)=2g(u,v)+3g(w,v)g(u,av+bw)=ag(u,v)+bg(u,w)
Consequently, in a given (arbitrary) coordinate system, its operation on two vectors can be written in terms of their components as a bilinear expression:
g ( u , v ) = g α β u α v β (implied summation on α , β ) = g 11 u 1 v 1 + g 12 u 1 v 2 + g 21 u 2 v 1 + . g ( u , v ) = g α β u α v β  (implied summation on  α , β ) = g 11 u 1 v 1 + g 12 u 1 v 2 + g 21 u 2 v 1 + . {:[g(u","v)=g_(alpha beta)u^(alpha)v^(beta)],[quad" (implied summation on "alpha","beta)],[=g_(11)u^(1)v^(1)+g_(12)u^(1)v^(2)+g_(21)u^(2)v^(1)+cdots.]:}\begin{aligned} \boldsymbol{g}(\boldsymbol{u}, \boldsymbol{v})= & g_{\alpha \beta} u^{\alpha} v^{\beta} \\ & \quad \text { (implied summation on } \alpha, \beta) \\ = & g_{11} u^{1} v^{1}+g_{12} u^{1} v^{2}+g_{21} u^{2} v^{1}+\cdots . \end{aligned}g(u,v)=gαβuαvβ (implied summation on α,β)=g11u1v1+g12u1v2+g21u2v1+.
The quantities g α β = g β α ( α g α β = g β α ( α g_(alpha beta)=g_(beta alpha)(alphag_{\alpha \beta}=g_{\beta \alpha}(\alphagαβ=gβα(α and β β beta\betaβ running from 0 to 3 in spacetime, from 1 to 2 on the apple) are called the "components of g g g\boldsymbol{g}g in the given coordinate system."
B. Components of the metric in local Lorentz and local Euclidean frames:
To connect the metric with our previous descriptions of the local geometry, introduce
local Euclidean coordinates (on apple) or local Lorentz coordinates (in spacetime).
Let ξ ξ xi\xiξ be the separation vector reaching from a a aaa to B B B\mathscr{B}B. Its components in the local Euclidean (Lorentz) coordinates are
ξ α = x α ( B ) x α ( A ) ξ α = x α ( B ) x α ( A ) xi^(alpha)=x^(alpha)(B)-x^(alpha)(A)\xi^{\alpha}=x^{\alpha}(\mathscr{B})-x^{\alpha}(\mathscr{A})ξα=xα(B)xα(A)
(cf. Box 1.1). Then the squared length of u G 3 u G 3 u_(G*3)\boldsymbol{u}_{G \cdot 3}uG3, which is the same as the squared distance from A A A\mathscr{A}A to B B B\mathscr{B}B, must be (cf. I.B. and II.B. above)
ξ ξ = g ( ξ , ξ ) = g α β ξ α ξ β = s a ! 0 2 = ( ξ 1 ) 2 + ( ξ 2 ) 2 on apple = ( ξ 0 ) 2 + ( ξ 1 ) 2 + ( ξ 2 ) 2 + ( ξ 3 ) 2 in spacetime. ξ ξ = g ( ξ , ξ ) = g α β ξ α ξ β = s a ! 0 2 = ξ 1 2 + ξ 2 2  on apple  = ξ 0 2 + ξ 1 2 + ξ 2 2 + ξ 3 2  in spacetime.  {:[xi*xi=g(xi","xi)=g_(alpha beta)xi^(alpha)xi^(beta)],[=s_(a!0^(2))=(xi^(1))^(2)+(xi^(2))^(2)" on apple "],[=-(xi^(0))^(2)+(xi^(1))^(2)+(xi^(2))^(2)+(xi^(3))^(2)],[" in spacetime. "]:}\begin{aligned} \boldsymbol{\xi} \cdot \boldsymbol{\xi} & =\boldsymbol{g}(\boldsymbol{\xi}, \boldsymbol{\xi})=g_{\alpha \beta} \xi^{\alpha} \xi^{\beta} \\ & =s_{a!0^{2}}=\left(\xi^{1}\right)^{2}+\left(\xi^{2}\right)^{2} \text { on apple } \\ & =-\left(\xi^{0}\right)^{2}+\left(\xi^{1}\right)^{2}+\left(\xi^{2}\right)^{2}+\left(\xi^{3}\right)^{2} \\ & \text { in spacetime. } \end{aligned}ξξ=g(ξ,ξ)=gαβξαξβ=sa!02=(ξ1)2+(ξ2)2 on apple =(ξ0)2+(ξ1)2+(ξ2)2+(ξ3)2 in spacetime. 
Consequently, the components of the metric are
g 11 = g 22 = 1 , g 12 = g 21 = 0 ; i.e., g α β = δ α β on apple, in local Euclidean coordinates; g 00 = 1 , g 0 k = 0 , g j k = δ j k in spacetime, in local Lorentz coordinates. g 11 = g 22 = 1 , g 12 = g 21 = 0 ;  i.e.,  g α β = δ α β  on apple, in   local Euclidean   coordinates;  g 00 = 1 , g 0 k = 0 , g j k = δ j k  in spacetime, in   local Lorentz   coordinates.  {:[g_(11)=g_(22)=1","g_(12)=g_(21)=0;],[" i.e., "g_(alpha beta)=delta_(alpha beta)" on apple, in "],[" local Euclidean "],[" coordinates; "],[g_(00)=-1","g_(0k)=0","g_(jk)=delta_(jk)],[" in spacetime, in "],[" local Lorentz "],[" coordinates. "]:}\begin{aligned} & g_{11}=g_{22}=1, g_{12}=g_{21}=0 ; \\ & \text { i.e., } g_{\alpha \beta}=\delta_{\alpha \beta} \text { on apple, in } \\ & \text { local Euclidean } \\ & \text { coordinates; } \\ & g_{00}=-1, g_{0 k}= 0, g_{j k}=\delta_{j k} \\ & \text { in spacetime, in } \\ & \text { local Lorentz } \\ & \text { coordinates. } \end{aligned}g11=g22=1,g12=g21=0; i.e., gαβ=δαβ on apple, in  local Euclidean  coordinates; g00=1,g0k=0,gjk=δjk in spacetime, in  local Lorentz  coordinates. 
These special components of the metric in local Lorentz coordinates are written here and hereafter as g α ^ β g α ^ β g_( hat(alpha)beta)g_{\hat{\alpha} \beta}gα^β or η α β η α β eta_(alpha beta)\eta_{\alpha \beta}ηαβ, by analogy with the Kronecker delta δ α β δ α β delta_(alpha beta)\delta_{\alpha \beta}δαβ. In matrix notation:
to empirical test in the appropriate, very special coordinate systems: Euclidean coordinates in Euclidean geometry; the natural generalization of Euclidean coordinates (local Lorentz coordinates; local inertial frame) in the local Lorentz geometry of physics. However, the theorems rise above all coordinate systems in their content. They refer to intervals or distances. Those distances no more call on coordinates for their definition in our day than they did in the time of Euclid. Points in the great pile of hay that is spacetime; and distances between these points: that is geometry! State them in the coordinate-free language or in the language of coordinates: they are the same (Box 1.3).

§ 1.5. TIME

Time is defined so that motion looks simple.
Time is awake when all things sleep. Time stands straight when all things fall. Time shuts in all and will not be shut. Is, was, and shall be are Time's children.
O Reasoning, be witness, be stable.
VYASA, the Mahabarata (ca. A.D. 400)
Relative to a local Lorentz frame, a free particle "moves in a straight line with uniform velocity." What "straight" means is clear enough in the model inertial reference frame illustrated in Figure 1.7. But where does the "uniform velocity" come in ? Or where does "velocity" show itself? There is not even one clock in the drawing!
A more fully developed model of a Lorentz reference frame will have not only holes, as in Fig. 1.7, but also clock-activated shutters over each hole. The projectile can reach its target only if it (1) travels through the correct region in space and (2) gets through that hole in the correct interval of time ("window in time"). How then is time defined? Time is defined so that motion looks simple!
No standard of time is more widely used than the day, the time from one high noon to the next. Take that as standard, however, and one will find every good clock or watch clashing with it, for a simple reason. The Earth spins on its axis and also revolves in orbit about the sun. The motion of the sun across the sky arises from neither effect alone, but from the two in combination, different in magnitude though they are. The fast angular velocity of the Earth on its axis (roughly 366.25 complete turns per year) is wonderfully uniform. Not so the apparent angular velocity of the sun about the center of the Earth (one turn per year). It is greater than average by 2 per cent when the Earth in its orbit (eccentricity 0.017 ) has come 1 per cent closer than average to the sun (Kepler's law) and lower by 2 per cent when the Earth is 1 per cent further than average from the sun. In the first case, the momentary rate of rotation of the sun across the sky, expressed in turns per year, is approximately
366.25 ( 1 + 0.02 ) 366.25 ( 1 + 0.02 ) 366.25-(1+0.02)366.25-(1+0.02)366.25(1+0.02)
The time coordinate of a local Lorentz frame is so defined that motion looks simple
in the other,
366.25 ( 1 0.02 ) 366.25 ( 1 0.02 ) 366.25-(1-0.02)366.25-(1-0.02)366.25(10.02)
Taking the "mean solar day" to contain 24 × 3 , 600 = 86 , 400 24 × 3 , 600 = 86 , 400 24 xx3,600=86,40024 \times 3,600=86,40024×3,600=86,400 standard seconds, one sees that, when the Earth is 1 per cent closer to (or further from) the sun than average, then the number of standard seconds from one high noon to the next is greater (or less) than normal by
0.02 (drop in turns per year) 365.25 (turns per year on average) 86 , 400 sec 4.7 sec . 0.02  (drop in turns per year)  365.25  (turns per year on average)  86 , 400 sec 4.7 sec . (0.02" (drop in turns per year) ")/(365.25" (turns per year on average) ")86,400sec∼4.7sec.\frac{0.02 \text { (drop in turns per year) }}{365.25 \text { (turns per year on average) }} 86,400 \mathrm{sec} \sim 4.7 \mathrm{sec} .0.02 (drop in turns per year) 365.25 (turns per year on average) 86,400sec4.7sec.
This is the bookkeeping on time from noon to noon. No standard of time that varies so much from one month to another is acceptable. If adopted, it would make the speed of light vary from month to month!
This lack of uniformity, once recognized (and it was already recognized by the ancients), forces one to abandon the solar day as the standard of time; that day does not make motion look simple. Turn to a new standard that eliminates the motion of the Earth around the sun and concentrates on the spin of the Earth about its axis: the sidereal day, the time between one arrival of a star at the zenith and the next arrival of that star at the zenith. Good! Or good, so long as one's precision of measurement does not allow one to see changes in the intrinsic angular velocity of the Earth. What clock was so bold as first to challenge the spin of the Earth for accuracy? The machinery of the heavens.
Halley (1693) and later others, including Kant (1754), suspected something was amiss from apparent discrepancies between the paths of totality in eclipses of the sun, as predicted by Newtonian gravitation theory using the standard of time then current, and the location of the sites where ancient Greeks and Romans actually recorded an eclipse on the day in question. The moon casts a moving shadow in space. On the day of a solar eclipse, that shadow paints onto the disk of the spinning Earth a black brush stroke, often thousands of kilometers in length, but of width generally much less than a hundred kilometers. He who spins the globe upon the table and wants to make the shadow fall rightly on it must calculate back meticulously to determine two key items: (1) where the moon is relative to Earth and sun at each moment on the ancient day in question; and (2) how much angle the Earth has turned through from then until now. Take the eclipse of Jan. 14, A.D. 484, as an example (Figure 1.8), and assume the same angular velocity for the Earth in the intervening fifteen centuries as the Earth had in 1900 (astronomical reference point). One comes out wrong. The Earth has to be set back by 30 30 30^(@)30^{\circ}30 (or the moon moved from its computed position, or some combination of the two effects) to make the Athens observer fall under the black brush. To catch up those 30 30 30^(@)30^{\circ}30 (or less, if part of the effect is due to a slow change in the angular momentum of the moon), the Earth had to turn faster in the past than it does today. Assigning most of the discrepancy to terrestrial spin-down (rate of spin-down compatible with modern atomic-clock evidence), and assuming a uniform rate of slowing from then to now
Figure 1.8.
Calculated path of totality for the eclipse of January 14, A.D. 484 (left; calculation based on no spin-down of Earth relative to its 1900 angular velocity) contrasted with the same path as set ahead enough to put the center of totality (at sunrise) at Athens [displacement very close to 30 30 30^(@)30^{\circ}30; actual figure of deceleration adopted in calculations, 32.75 arc sec/(century) 2 2 ^(2){ }^{2}2 ]. This is "undoubtedly the most reliable of all ancient European eclipses," according to Dr. F. R. Stephenson, of the Department of Geophysics and Planetary Physics of the University of Newcastle upon Tyne, who most kindly prepared this diagram especially for this book. He has also sent a passage from the original Greek biography of Proclus of Athens (died at Athens A.D. 485) by Marinus of Naples, reading, "Nor were there portents wanting in the year which preceded his death; for example, such a great eclipse of the Sun that night seemed to fall by day. For a profound darkness arose so that stars even appeared in the sky. This happened in the eastern sky when the Sun dwelt in Capricorn" [from Westermann and Boissonade (1878)].
Does this 30 30 30^(@)30^{\circ}30 for this eclipse, together with corresponding amounts for other eclipses, represent the "right" correction? "Right" is no easy word. From one total eclipse of the sun in the Mediterranean area to another is normally many years. The various provinces of the Greek and Roman worlds were far from having a uniform level of peace and settled life, and even farther from having a uniform standard of what it is to observe an eclipse and put it down for posterity. If the scores of records of the past are unhappily fragmentary, even more unhappy has been the willingness of a few uncritical "investigators" in recent times to rush in and identify this and that historical event with this and that calculated eclipse. Fortunately, by now a great literature is available on the secular deceleration of the Earth's rotation, in the highest tradition of critical scholarship, both astronomical and historical. In addition to the books of O. Neugebauer (1959) and Munk and MacDonald (1960), the paper of Curott (1966), and items cited by these workers, the following are key items. (For direction to them, we thank Professor Otto Neuge-bauer-no relation to the other Neugebauer cited below!) For the ancient records, and for calculations of the tracks of ancient eclipses, F. K. Ginzel (1882, 1883, 1884); for an atlas of calculated eclipse tracks, Oppolzer (1887) and Ginzel (1899); and for a critical analysis of the evidence. P. V. Neugebauer (1927, 1929, and 1930). This particular eclipse was chosen rather than any other because of the great reliability of the historical record of it.
(angular velocity correction proportional to first power of elapsed time: angle correction itself proportional to square of elapsed time), one estimates from a correction of
30 or 2 hours 1 , 500 years ago 30  or  2  hours  1 , 500  years ago  30^(@)" or "2" hours "quad1,500" years ago "30^{\circ} \text { or } 2 \text { hours } \quad 1,500 \text { years ago }30 or 2 hours 1,500 years ago 
the following corrections for intermediate times:
30 / 10 2 , or 1.2 min 150 years ago, 30 / 10 4 , or 0.8 sec 15 years ago. 30 / 10 2 , or  1.2 min 150  years ago,  30 / 10 4 , or  0.8 sec 15  years ago.  {:[30^(@)//10^(2)", or "1.2min,150" years ago, "],[30^(@)//10^(4)", or "0.8sec,15" years ago. "]:}\begin{array}{lr} 30^{\circ} / 10^{2} \text {, or } 1.2 \mathrm{~min} & 150 \text { years ago, } \\ 30^{\circ} / 10^{4} \text {, or } 0.8 \mathrm{sec} & 15 \text { years ago. } \end{array}30/102, or 1.2 min150 years ago, 30/104, or 0.8sec15 years ago. 
Thus one sees the downfall of the Earth as a standard of time and its replacement by the orbital motions of the heavenly bodies as a better standard: a standard that does more to "make motion look simple." Astronomical time is itself in turn today being supplanted by atomic time as a standard of reference (see Box 1.4, "Time Today").
Look at a bad clock for a good view of how time is defined. Let t t ttt be time on a "good" clock (time coordinate of a local inertial frame); it makes the tracks of free particles through the local region of spacetime look straight. Let T ( t ) T ( t ) T(t)T(t)T(t) be the reading of the "bad" clock; it makes the world lines of free particles through the local region of spacetime look curved (Figure 1.9). The old value of the acceleration, translated into the new ("bad") time, becomes
0 = d 2 x d t 2 = d d t ( d T d t d x d T ) = d 2 T d t 2 d x d T + ( d T d t ) 2 d 2 x d T 2 0 = d 2 x d t 2 = d d t d T d t d x d T = d 2 T d t 2 d x d T + d T d t 2 d 2 x d T 2 0=(d^(2)x)/(dt^(2))=(d)/(dt)((dT)/(dt)(dx)/(dT))=(d^(2)T)/(dt^(2))(dx)/(dT)+((dT)/(dt))^(2)(d^(2)x)/(dT^(2))0=\frac{d^{2} x}{d t^{2}}=\frac{d}{d t}\left(\frac{d T}{d t} \frac{d x}{d T}\right)=\frac{d^{2} T}{d t^{2}} \frac{d x}{d T}+\left(\frac{d T}{d t}\right)^{2} \frac{d^{2} x}{d T^{2}}0=d2xdt2=ddt(dTdtdxdT)=d2Tdt2dxdT+(dTdt)2d2xdT2
To explain the apparent accelerations of the particles, the user of the new time introduces a force that one knows to be fictitious:
(1.2) F x = m d 2 x d T 2 = m ( d x d T ) ( d 2 T d t 2 ) ( d T d t ) 2 (1.2) F x = m d 2 x d T 2 = m d x d T d 2 T d t 2 d T d t 2 {:(1.2)F_(x)=m(d^(2)x)/(dT^(2))=-m(((dx)/(dT))((d^(2)T)/(dt^(2))))/(((dT)/(dt))^(2)):}\begin{equation*} F_{x}=m \frac{d^{2} x}{d T^{2}}=-m \frac{\left(\frac{d x}{d T}\right)\left(\frac{d^{2} T}{d t^{2}}\right)}{\left(\frac{d T}{d t}\right)^{2}} \tag{1.2} \end{equation*}(1.2)Fx=md2xdT2=m(dxdT)(d2Tdt2)(dTdt)2
It is clear from this example of a "bad" time that Newton thought of a "good" time when he set up the principle that "Time flows uniformly" ( d 2 T / d t 2 = 0 ) d 2 T / d t 2 = 0 (d^(2)T//dt^(2)=0)\left(d^{2} T / d t^{2}=0\right)(d2T/dt2=0). Time is defined to make motion look simple!
The principle of uniformity, taken by itself, leaves free the scale of the time variable. The quantity T = a t + b T = a t + b T=at+bT=a t+bT=at+b satisfies the requirement as well as t t ttt itself. The history of timekeeping discloses many choices of the unit and origin of time. Each one required some human action to give it sanction, from the fiat of a Pharaoh to the communique of a committee. In this book the amount of time it takes light to travel one centimeter is decreed to be the unit of time. Spacelike intervals and timelike intervals are measured in terms of one and the same geometric unit: the centimeter. Any other decision would complicate in analysis what is simple in nature. No other choice would live up to Minkowski's words, "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."
Figure 1.9.
Good clock (left) vs. bad clock (right) as seen in the maps they give of the same free particles moving through the same region of spacetime. The world lines as depicted at the right give the impression that a force is at work. The good definition of time eliminates such fictitious forces. The dashed lines connect corresponding instants on the two time scales.
One can measure time more accurately today than distance. Is that an argument against taking the elementary unit to be the centimeter? No, provided that this definition of the centimeter is accepted: the geometrodynamic standard centimeter is the fraction
(1.3) 1 / ( 9.460546 × 10 17 ) (1.3) 1 / 9.460546 × 10 17 {:(1.3)1//(9.460546 xx10^(17)):}\begin{equation*} 1 /\left(9.460546 \times 10^{17}\right) \tag{1.3} \end{equation*}(1.3)1/(9.460546×1017)
of the interval between the two "effective equinoxes" that bound the tropical year 1900.0. The tropical year 1900.0 has already been recognized internationally as the fiducial interval by reason of its definiteness and the precision with which it is known. Standards committees have defined the ephemeris second so that 31 , 556 , 925.974 sec 31 , 556 , 925.974 sec 31,556,925.974sec31,556,925.974 \mathrm{sec}31,556,925.974sec make up that standard interval. Were the speed of light known with perfect precision, the standards committees could have given in the same breath the number of centimeters in the standard interval. But it isn't; it is known to only six decimals. Moreover, the international centimeter is defined in terms of the orange-red wavelength of Kr 86 Kr 86 Kr^(86)\mathrm{Kr}^{86}Kr86 to only nine decimals ( 16 , 507.6373 16 , 507.6373 16,507.637316,507.637316,507.6373 wavelengths). Yet the standard second is given to 11 decimals. We match the standard second by arbitrarily defining the geometrodynamic standard centimeter so that
9.4605460000 × 10 17 9.4605460000 × 10 17 9.4605460000 xx10^(17)9.4605460000 \times 10^{17}9.4605460000×1017
such centimeters are contained in the standard tropical year 1900.0. The speed of light then becomes exactly
(1.4) 9.4605460000 × 10 17 31 , 556 , 925.974 geometrodynamic cm / sec . (1.4) 9.4605460000 × 10 17 31 , 556 , 925.974  geometrodynamic  cm / sec . {:(1.4)(9.4605460000 xx10^(17))/(31,556,925.974)" geometrodynamic "cm//sec.:}\begin{equation*} \frac{9.4605460000 \times 10^{17}}{31,556,925.974} \text { geometrodynamic } \mathrm{cm} / \mathrm{sec} . \tag{1.4} \end{equation*}(1.4)9.4605460000×101731,556,925.974 geometrodynamic cm/sec.
This is compatible with the speed of light, as known in 1967, in units of "international cm / sec cm / sec cm//sec\mathrm{cm} / \mathrm{sec}cm/sec ":
29 , 979 , 300 , 000 ± 30 , 000 29 , 979 , 300 , 000 ± 30 , 000 29,979,300,000+-30,00029,979,300,000 \pm 30,00029,979,300,000±30,000 international cm / sec cm / sec cm//sec\mathrm{cm} / \mathrm{sec}cm/sec.
Box 1.4 TIME TODAY
Prior to 1956 the second was defined as the fraction 1 / 86 , 400 1 / 86 , 400 1//86,4001 / 86,4001/86,400 of the mean solar day.
From 1956 to 1967 the "second" meant the ephemeris second, defined as the fraction 1 / ( 31 , 556 , 925.9747 ) 1 / ( 31 , 556 , 925.9747 ) 1//(31,556,925.9747)1 /(31,556,925.9747)1/(31,556,925.9747) of the tropical year 00 h 00 m 00 s December 31, 1899.
Since 1967 the standard second has been the SI (Système International) second, defined as 9 , 192 , 631 , 770 9 , 192 , 631 , 770 9,192,631,7709,192,631,7709,192,631,770 periods of the unperturbed microwave transition between the two hyperfine levels of the ground state of Cs 133 Cs 133 Cs^(133)\mathrm{Cs}^{133}Cs133.
Like the foregoing evolution of the unit for the time interval, the evolution of a time coordinate has been marked by several stages.
Universal time, UTO, is based on the count of days as they actually occurred historically; in other words, on the actual spin of the earth on its axis; historically, on mean solar time (solar position as corrected by the "equation of time"; i.e., the faster travel of the earth when near the sun than when far from the sun) as determined at Greenwich Observatory.
UT1, the "navigator's time scale," is the same time as corrected for the wobble of the earth on its axis ( Δ t 0.05 sec Δ t 0.05 sec Delta t∼0.05sec\Delta t \sim 0.05 \mathrm{sec}Δt0.05sec ).
UT2 is UT1 as corrected for the periodic fluctuations of unknown origin with periods of onehalf year and one year ( Δ t 0.05 sec Δ t 0.05 sec Delta t∼0.05sec\Delta t \sim 0.05 \mathrm{sec}Δt0.05sec; measured to 3 ms in one day).
Ephemeris Time, ET (as defined by the theory of gravitation and by astronomical observations and calculations), is essentially determined by the orbital motion of the earth around the sun. "Measurement uncertainties limit the realization of accurate ephemeris time to about 0.05 sec for a nine-year average."
Coordinated Universal Time (UTC) is broadcast on stations such as WWV. It was adopted internationally in February 1971 to become effective January 1,1972 . The clock rate is controlled by atomic clocks to be as uniform as possible for one year (atomic time is measured to 0.1 0.1 ∼0.1\sim 0.10.1 microsec in 1 min , with diffusion rates of 0.1 microsec per day for ensembles of clocks), but is changed by the infrequent addition or deletion of a second-called a "leap second"-so that UTC never differs more than 0.7 sec from the navigator's time scale, UT1.

THE TIMES

Wednesday
June 211972

Time suspended for a second

Time will stand still throughout the world for one second at midnight, June 30. All radio time signals will insert a "leap second" to bring Greenwich Mean Time into line with the earth's loss of three thousandths of a second a day.
The signal from the Royal Greenwich Observatory to Broadcasting House at midnight GMT (1 am BST July 1) will be six short pips marking the seconds 55 to 60 inclusive, followed by a lengthened signal at the following second to mark the new minute.
The foregoing account is abstracted from J. A. Barnes (1971). The following is extracted from a table (not official at time of receipt), kindly supplied by the Time and Frequency Division of the U.S. National Bureau of Standards in Boulder, Colorado.
Timekeeping capabilities of some familiar clocks are as follows:
Tuning fork wrist watch (1960), 1 min / mo 1 min / mo 1min//mo1 \mathrm{~min} / \mathrm{mo}1 min/mo.
Quartz crystal clock (1921-1930), 1 μ sec / 1 μ sec / 1musec//1 \mu \mathrm{sec} /1μsec/ day, 1 sec / yr 1 sec / yr 1sec//yr1 \mathrm{sec} / \mathrm{yr}1sec/yr.
Quartz crystal wrist watch (1971), 0.2 sec / 2 0.2 sec / 2 0.2sec//20.2 \mathrm{sec} / 20.2sec/2 mos., 1 sec / yr 1 sec / yr 1sec//yr1 \mathrm{sec} / \mathrm{yr}1sec/yr.
Cesium beam (atomic resonance, Cs 133 Cs 133 Cs^(133)\mathrm{Cs}^{133}Cs133 ), (19521955),
0.1 μ sec / 0.1 μ sec / 0.1 musec//0.1 \mu \mathrm{sec} /0.1μsec/ day, 0.5 μ sec / mo 0.5 μ sec / mo 0.5 musec//mo0.5 \mu \mathrm{sec} / \mathrm{mo}0.5μsec/mo.
Rubidium gas cell ( Rb 87 Rb 87 Rb^(87)\mathrm{Rb}^{87}Rb87 resonance), (1957), 0.1 μ sec / 0.1 μ sec / 0.1 musec//0.1 \mu \mathrm{sec} /0.1μsec/ day, 1 5 μ sec / mo 1 5 μ sec / mo 1-5musec//mo1-5 \mu \mathrm{sec} / \mathrm{mo}15μsec/mo.
Hydrogen maser (1960), 0.01 μ sec / 2 hr 0.01 μ sec / 2 hr 0.01 musec//2hr0.01 \mu \mathrm{sec} / 2 \mathrm{hr}0.01μsec/2hr, 0.1 μ sec / 0.1 μ sec / 0.1 musec//0.1 \mu \mathrm{sec} /0.1μsec/ day.
Methane stabilized laser (1969), 0.01 μ sec / 100 sec 0.01 μ sec / 100 sec 0.01 musec//100sec0.01 \mu \mathrm{sec} / 100 \mathrm{sec}0.01μsec/100sec.
Recent measurements [Evenson et al. (1972)] change the details of the foregoing 1967 argument, but not the principles.

§1.6. CURVATURE

Gravitation seems to have disappeared. Everywhere the geometry of spacetime is locally Lorentzian. And in Lorentz geometry, particles move in a straight line with constant velocity. Where is any gravitational deflection to be seen in that? For answer, turn back to the apple (Figure 1.1). Inspect again the geodesic tracks of the ants on the surface of the apple. Note the reconvergence of two nearby geodesics that originally diverged from a common point. What is the analog in the real world of physics? What analogous concept fits Einstein's injunction that physics is only simple when analyzed locally? Don't look at the distance from the spaceship to the Earth. Look at the distance from the spaceship to a nearby spaceship! Or, to avoid any possible concern about attraction between the two ships, look at two nearby test particles in orbit about the Earth. To avoid distraction by the nonlocal element (the Earth) in the situation, conduct the study in the interior of a spaceship, also in orbit about the Earth. But this region has already been counted as a local inertial frame! What gravitational physics is to be seen there? None. Relative to the spaceship and therefore relative to each other, the two test particles move in a straight line with uniform velocity, to the precision of measurement that is contemplated (see Box 1.5, "Test for Flatness"). Now the key point begins to appear: precision of measurement. Increase it until one begins to discern the gradual acceleration of the test particles away from each other, if they lie along a common radius through the center of the Earth; or toward each other, if their separation lies perpendicular to that line. In Newtonian language, the source of these accelerations is the tide-producing action of the Earth. To the observer in the spaceship, however, no Earth is to be seen. And following Einstein, he knows it is important to analyze motion locally. He represents the separation of the new test particle from the fiducial test particle by the vector ξ k ( k = 1 , 2 , 3 ξ k ( k = 1 , 2 , 3 xi^(k)(k=1,2,3\xi^{k}(k=1,2,3ξk(k=1,2,3; components measured in a local Lorentz frame). For the acceleration of this separation, one knows from Newtonian physics what he will find: if the Cartesian z z zzz-axis is in the radial direction, then
d 2 ξ x d t 2 = G m conv c 2 r 3 ξ x ; (1.5) d 2 ξ y d t 2 = G m conv c 2 r 3 ξ y ; d 2 ξ z d t 2 = 2 G m conv ξ 2 r 3 c z . d 2 ξ x d t 2 = G m conv  c 2 r 3 ξ x ; (1.5) d 2 ξ y d t 2 = G m conv  c 2 r 3 ξ y ; d 2 ξ z d t 2 = 2 G m conv  ξ 2 r 3 c z . {:[(d^(2)xi^(x))/(dt^(2))=-(Gm_("conv "))/(c^(2)r^(3))xi^(x);],[(1.5)(d^(2)xi^(y))/(dt^(2))=-(Gm_("conv "))/(c^(2)r^(3))xi^(y);],[(d^(2)xi^(z))/(dt^(2))=(2Gm_("conv ")xi^(2)r^(3))/(c^(z)).]:}\begin{align*} & \frac{d^{2} \xi^{x}}{d t^{2}}=-\frac{G m_{\text {conv }}}{c^{2} r^{3}} \xi^{x} ; \\ & \frac{d^{2} \xi^{y}}{d t^{2}}=-\frac{G m_{\text {conv }}}{c^{2} r^{3}} \xi^{y} ; \tag{1.5}\\ & \frac{d^{2} \xi^{z}}{d t^{2}}=\frac{2 G m_{\text {conv }} \xi^{2} r^{3}}{c^{z}} . \end{align*}d2ξxdt2=Gmconv c2r3ξx;(1.5)d2ξydt2=Gmconv c2r3ξy;d2ξzdt2=2Gmconv ξ2r3cz.
Proof: In Newtonian physics the acceleration of a single particle toward the center of the Earth in conventional units of time is G m conv / r 2 G m conv  / r 2 Gm_("conv ")//r^(2)G m_{\text {conv }} / r^{2}Gmconv /r2, where G G GGG is the Newtonian constant of gravitation, 6.670 × 10 8 cm 3 / g sec 2 6.670 × 10 8 cm 3 / g sec 2 6.670 xx10^(-8)cm^(3)//gsec^(2)6.670 \times 10^{-8} \mathrm{~cm}^{3} / \mathrm{g} \mathrm{sec}^{2}6.670×108 cm3/gsec2 and m conv m conv  m_("conv ")m_{\text {conv }}mconv  is the mass of the Earth in conventional units of grams. In geometric units of time ( cm of light-travel time),
Gravitation is manifest in relative acceleration of neighboring test particles
Relative acceleration is caused by curvature
the acceleration is G m conv / c 2 r 2 G m conv  / c 2 r 2 Gm_("conv ")//c^(2)r^(2)G m_{\text {conv }} / c^{2} r^{2}Gmconv /c2r2. When the two particles are separated by a distance ξ ξ xi\xiξ perpendicular to r r rrr, the one downward acceleration vector is out of line with the other by the angle ξ / r ξ / r xi//r\xi / rξ/r. Consequently one particle accelerates toward the other by the stated amount. When the separation is parallel to r r rrr, the relative acceleration is given by evaluating the Newtonian acceleration at r r rrr and at r + ξ r + ξ r+xir+\xir+ξ, and taking the difference ( ξ ξ xi\xiξ times d / d r d / d r d//drd / d rd/dr ) Q.E.D. In conclusion, the "local tide-producing acceleration" of Newtonian gravitation theory provides the local description of gravitation that Einstein bids one to seek.
What has this tide-producing acceleration to do with curvature? (See Box 1.6.) Look again at the apple or, better, at a sphere of radius a a aaa (Figure 1.10). The separation of nearby geodesics satisfies the "equation of geodesic deviation,"
(1.6) d 2 ξ / d s 2 + R ξ = 0 (1.6) d 2 ξ / d s 2 + R ξ = 0 {:(1.6)d^(2)xi//ds^(2)+R xi=0:}\begin{equation*} d^{2} \xi / d s^{2}+R \xi=0 \tag{1.6} \end{equation*}(1.6)d2ξ/ds2+Rξ=0
Here R = 1 / a 2 R = 1 / a 2 R=1//a^(2)R=1 / a^{2}R=1/a2 is the so-called Gaussian curvature of the surface. For the surface of the apple, the same equation applies, with the one difference that the curvature R R RRR varies from place to place.

Box 1.5 TEST FOR FLATNESS

  1. Specify the extension in space L L LLL ( cm or m ) and extension in time T T TTT ( cm or m of light travel time) of the region under study.
  2. Specify the precision δ ξ δ ξ delta xi\delta \xiδξ with which one can measure the separation of test particles in this region.
  3. Follow the motion of test particles moving along initially parallel world lines through this region of spacetime.
  4. When the world lines remain parallel to the precision δ ξ δ ξ delta xi\delta \xiδξ for all directions of travel, then one says that "in a region so limited and to a precision so specified, spacetime is flat."
    example: Region just above the surface of the earth, 100 m × 100 m × 100 m 100 m × 100 m × 100 m 100mxx100mxx100m100 \mathrm{~m} \times 100 \mathrm{~m} \times 100 \mathrm{~m}100 m×100 m×100 m (space extension), followed for 10 9 m 10 9 m 10^(9)m10^{9} \mathrm{~m}109 m of light-travel time ( T conv T conv  T_("conv ")∼T_{\text {conv }} \simTconv  3 sec ). Mass of Earth, m conv = 5.98 × 10 27 g m conv  = 5.98 × 10 27 g m_("conv ")=5.98 xx10^(27)gm_{\text {conv }}=5.98 \times 10^{27} \mathrm{~g}mconv =5.98×1027 g, m = ( 0.742 × 10 28 cm / g ) × ( 5.98 × 10 27 g ) = m = 0.742 × 10 28 cm / g × 5.98 × 10 27 g = m=(0.742 xx10^(-28)(cm)//g)xx(5.98 xx10^(27)(g))=m=\left(0.742 \times 10^{-28} \mathrm{~cm} / \mathrm{g}\right) \times\left(5.98 \times 10^{27} \mathrm{~g}\right)=m=(0.742×1028 cm/g)×(5.98×1027 g)= 0.444 cm [see eq. (1.12)]. Tide-producing acceleration R z 0 z 0 R z 0 z 0 R^(z)_(0z0)R^{z}{ }_{0 z 0}Rz0z0 (relative acceleration in z z zzz-direction of two test particles initially at rest and separated from each other by 1 cm of vertical elevation) is
( d / d r ) ( m / r 2 ) = 2 m / r 3 = 0.888 cm / ( 6.37 × 10 8 cm ) 3 = 3.44 × 10 27 cm 2 ( d / d r ) m / r 2 = 2 m / r 3 = 0.888 cm / 6.37 × 10 8 cm 3 = 3.44 × 10 27 cm 2 {:[(d//dr)(m//r^(2))=-2m//r^(3)],[=-0.888cm//(6.37 xx10^(8)(cm))^(3)],[=-3.44 xx10^(-27)cm^(-2)]:}\begin{aligned} (d / d r)\left(m / r^{2}\right) & =-2 m / r^{3} \\ & =-0.888 \mathrm{~cm} /\left(6.37 \times 10^{8} \mathrm{~cm}\right)^{3} \\ & =-3.44 \times 10^{-27} \mathrm{~cm}^{-2} \end{aligned}(d/dr)(m/r2)=2m/r3=0.888 cm/(6.37×108 cm)3=3.44×1027 cm2
("cm of relative displacement per cm of lighttravel time per cm of light-travel time per cm of vertical separation"). Two test particles with a vertical separation ξ z = 10 4 cm ξ z = 10 4 cm xi^(z)=10^(4)cm\xi^{z}=10^{4} \mathrm{~cm}ξz=104 cm acquire in the time t = 10 11 cm t = 10 11 cm t=10^(11)cmt=10^{11} \mathrm{~cm}t=1011 cm (difference between time and proper time negligible for such slowly moving test particles) a relative displacement
δ ξ z = 1 2 R 0 z 0 z t 2 ξ z = 1.72 × 10 27 cm 2 ( 10 11 cm ) 2 10 4 cm = 1.72 mm . δ ξ z = 1 2 R 0 z 0 z t 2 ξ z = 1.72 × 10 27 cm 2 10 11 cm 2 10 4 cm = 1.72 mm . {:[deltaxi^(z)=-(1)/(2)R_(0z0^(z)t^(2)xi^(z))],[=1.72 xx10^(-27)cm^(-2)(10^(11)(cm))^(2)10^(4)cm],[=1.72mm.]:}\begin{aligned} \delta \xi^{z} & =-\frac{1}{2} R_{0 z 0^{z} t^{2} \xi^{z}} \\ & =1.72 \times 10^{-27} \mathrm{~cm}^{-2}\left(10^{11} \mathrm{~cm}\right)^{2} 10^{4} \mathrm{~cm} \\ & =1.72 \mathrm{~mm} . \end{aligned}δξz=12R0z0zt2ξz=1.72×1027 cm2(1011 cm)2104 cm=1.72 mm.
(Change in relative separation less for other directions of motion). When the minimum uncertainty δ ξ δ ξ delta xi\delta \xiδξ attainable in a measurement over a 100 m spacing is "worse" than this figure (exceeds 1.72 mm ), then to this level of precision the region of spacetime under consideration can be treated as flat. When the uncertainty in measurement is "better" (less) than 1.72 mm , then one must limit attention to a smaller region of space or a shorter interval of time or both, to find a region of spacetime that can be regarded as flat to that precision.
Figure 1.10.
Curvature as manifested in the "acceleration of the separation" of two nearby geodesics. Two geodesics, originally parallel, and separated by the distance ("geodesic deviation") ξ 0 ξ 0 xi_(0)\xi_{0}ξ0, are no longer parallel when followed a distance s s sss. The separation is ξ = ξ 0 cos ϕ = ξ 0 cos ( s / a ) ξ = ξ 0 cos ϕ = ξ 0 cos ( s / a ) xi=xi_(0)cos phi=xi_(0)cos(s//a)\xi=\xi_{0} \cos \phi=\xi_{0} \cos (s / a)ξ=ξ0cosϕ=ξ0cos(s/a), where a a aaa is the radius of the sphere. The separation follows the equation of simple harmonic motion, d 2 ξ / d s 2 + ( 1 / a 2 ) ξ = 0 d 2 ξ / d s 2 + 1 / a 2 ξ = 0 d^(2)xi//ds^(2)+(1//a^(2))xi=0d^{2} \xi / d s^{2}+\left(1 / a^{2}\right) \xi=0d2ξ/ds2+(1/a2)ξ=0 ("equation of geodesic deviation").
The direction of the separation vector, ξ ξ xi\boldsymbol{\xi}ξ, is fixed fully by its orthogonality to the fiducial geodesic. Hence, no reference to the direction of ξ ξ xi\xiξ is needed or used in the equation of geodesic deviation; only the magnitude ξ ξ xi\xiξ of ξ ξ xi\xiξ appears there, and only the magnitude, not direction, of the relative acceleration appears.
In a space of more than two dimensions, an equation of the same general form applies, with several differences. In two dimensions the direction of acceleration of one geodesic relative to a nearby, fiducial geodesic is fixed uniquely by the demand that their separation vector, ξ ξ xi\boldsymbol{\xi}ξ, be perpendicular to the fiducial geodesic (see Figure 1.10). Not so in three dimensions or higher. There ξ ξ xi\xiξ can remain perpendicular to the fiducial geodesic but rotate about it (Figure 1.11). Thus, to specify the relative acceleration uniquely, one must give not only its magnitude, but also its direction.
The relative acceleration in three dimensions and higher, then, is a vector. Call it " D 2 ξ / d s 2 D 2 ξ / d s 2 D^(2)xi//ds^(2)D^{2} \xi / d s^{2}D2ξ/ds2," and call its four components " D 2 ξ α / d s 2 D 2 ξ α / d s 2 D^(2)xi^(alpha)//ds^(2)D^{2} \xi^{\alpha} / d s^{2}D2ξα/ds2." Why the capital D D DDD ? Why not " d 2 ξ α / d s 2 d 2 ξ α / d s 2 d^(2)xi^(alpha)//ds^(2)d^{2} \xi^{\alpha} / d s^{2}d2ξα/ds2 "? ? Because our coordinate system is completely arbitrary (cf. §1.2). The twisting and turning of the coordinate lines can induce changes from point to point in the components ξ α ξ α xi^(alpha)\xi^{\alpha}ξα of ξ ξ xi\boldsymbol{\xi}ξ, even if the vector ξ ξ xi\boldsymbol{\xi}ξ is not changing at all. Consequently, the accelerations of the components d 2 ξ α / d s 2 d 2 ξ α / d s 2 d^(2)xi^(alpha)//ds^(2)d^{2} \xi^{\alpha} / d s^{2}d2ξα/ds2 are generally not equal to the components D 2 ξ α / d s 2 D 2 ξ α / d s 2 D^(2xi alpha)//ds^(2)D^{2 \xi \alpha} / d s^{2}D2ξα/ds2 of the acceleration!
How, then, in curved spacetime can one determine the components D 2 ξ α / d s 2 D 2 ξ α / d s 2 D^(2xi alpha)//ds^(2)D^{2 \xi \alpha} / d s^{2}D2ξα/ds2 of the relative acceleration? By a more complicated version of the equation of geodesic deviation (1.6). Differential geometry (Part III of this book) provides us with a geometrical object called the Riemann curvature tensor, "Riemann." Riemann is
Curvature is characterized by Riemann tensor
Figure 1.11.
The separation vector ξ ξ xi\xiξ between two geodesics in a curved threedimensional manifold. Here ξ ξ xi\xiξ can not only change its length from point to point, but also rotate at a varying rate about the fiducial geodesic. Consequently, the relative acceleration of the geodesics must be characterized by a direction as well as a magnitude; it must be a vector, D 2 ξ / d s 2 D 2 ξ / d s 2 D^(2)xi//ds^(2)D^{2} \xi / d s^{2}D2ξ/ds2.

Box 1.6 CURVATURE OF WHAT?

Nothing seems more attractive at first glance than the idea that gravitation is a manifestation of the curvature of space (A), and nothing more ridiculous at a second glance (B). How can the tracks of a ball and of a bullet be curved so differently if that curvature arises from the geometry of space? No wonder that great Riemann did not give the world a geometric theory of gravity. Yes, at the age of 28 (June 10, 1854) he gave the world the mathematical machinery to define and calculate curvature (metric and Riemannian geometry). Yes, he spent his dying days at 40 working to find a unified account of electricity and gravitation. But if there was one reason more than any other why he failed to make the decisive connection between gravitation and curvature, it was this, that he thought of space and the curvature of space, not
of spacetime and the curvature of spacetime. To make that forward step took the forty years to special relativity (1905: time on the same footing as space) and then another ten years (1915: general relativity). Depicted in spacetime (C), the tracks of ball and bullet appear to have comparable curvature. In fact, however, neither track has any curvature at all. They both look curved in (C) only because one has forgotten that the spacetime they reside in is itself curved-curved precisely enough to make these tracks the straightest lines in existence ("geodesics").
If it is at first satisfying to see curvature, and curvature of spacetime at that, coming to the fore in so direct a way, then a little more reflection produces a renewed sense of concern. Curvature with respect to what? Not with respect to the labo-
quence of the curvature of space near the sun. Ray of light pursues geodesic, but geometry in which it travels is curved (actual travel takes place in spacetime rather than space; correct deflection is twice that given by above elementary picture). Deflection inversely proportional to angular separation between star and center of sun. See Box 40.1 for actual deflections observed at time of an eclipse.
ratory. The earth-bound laboratory has no simple status whatsoever in a proper discussion. First, it is no Lorentz frame. Second, even to mention the earth makes one think of an action-at-a-distance version of gravity (distance from center of earth to ball or bullet). In contrast, it was the whole point of Einstein that physics looks simple only when analyzed locally. To look at local physics, however, means to compare one geodesic of one test particle with geodesics of other test particles traveling (1) nearby with (2) nearly the same directions and (3) nearly the same speeds. Then one can "look at the separations between these nearby test particles and from the second time-rate of change of these separations and the equation of geodesic deviation' (equation 1.8) read out the curvature of spacetime."

B. Tracks of ball and bullet through space as seen in laboratory have very different curvatures.

C. Tracks of ball and bullet through spacetime, as recorded in laboratory, have comparable curvatures. Track compared to arc of circle: (radius) = = === (horizontal distance) ) 2 / 8 ) 2 / 8 )^(2)//8)^{2} / 8)2/8 (rise).
the higher-dimensional analog of the Gaussian curvature R R RRR of our apple's surface. Riemann is the mathematical embodiment of the bends and warps in spacetime. And Riemann is the agent by which those bends and warps (curvature of spacetime) produce the relative acceleration of geodesics.
Riemann, like the metric tensor g g g\boldsymbol{g}g of Box 1.3, can be thought of as a family of machines, one machine residing at each event in spacetime. Each machine has three slots for the insertion of three vectors:
Choose a fiducial geodesic (free-particle world line) passing through an event Q Q Q\mathscr{Q}Q, and denote its unit tangent vector (particle 4 -velocity) there by
(1.7) u = d x / d τ ; components, u α = d x α / d τ (1.7) u = d x / d τ ;  components,  u α = d x α / d τ {:(1.7)u=dx//d tau;" components, "u^(alpha)=dx^(alpha)//d tau:}\begin{equation*} \boldsymbol{u}=\boldsymbol{d} \boldsymbol{x} / d \tau ; \text { components, } u^{\alpha}=d x^{\alpha} / d \tau \tag{1.7} \end{equation*}(1.7)u=dx/dτ; components, uα=dxα/dτ
Choose another, neighboring geodesic, and denote by ξ ξ xi\xiξ its perpendicular separation from the fiducial geodesic. Then insert u u u\boldsymbol{u}u into the first slot of Riemann at Q , ξ Q , ξ Q,xi\mathcal{Q}, \boldsymbol{\xi}Q,ξ into the second slot, and u u u\boldsymbol{u}u into the third. Riemann will grind for awhile; then out will pop a new vector,

Riemann (u, ξ , u ) ξ , u ) xi,u)\boldsymbol{\xi}, \boldsymbol{u})ξ,u).

The equation of geodesic deviation states that this new vector is the negative of the relative acceleration of the two geodesics:
(1.8) D 2 ξ / d τ 2 + Riemann ( u , ξ , u ) = 0 (1.8) D 2 ξ / d τ 2 + Riemann ( u , ξ , u ) = 0 {:(1.8)D^(2)xi//dtau^(2)+Riemann(u","xi","u)=0:}\begin{equation*} D^{2} \boldsymbol{\xi} / d \tau^{2}+\operatorname{Riemann}(\boldsymbol{u}, \boldsymbol{\xi}, \boldsymbol{u})=0 \tag{1.8} \end{equation*}(1.8)D2ξ/dτ2+Riemann(u,ξ,u)=0
The Riemann tensor, like the metric tensor (Box 1.3), and like all other tensors, is a linear machine. The vector it puts out is a linear function of each vector inserted into a slot:
R i e m a n n ( 2 u , a w + b v , 3 r ) (1.9) = 2 × a × 3 Riemann ( u , w , r ) + 2 × b × 3 Riemann ( u , v , r ) . R i e m a n n ( 2 u , a w + b v , 3 r ) (1.9) = 2 × a × 3  Riemann  ( u , w , r ) + 2 × b × 3  Riemann  ( u , v , r ) . {:[Riemann(2u","aw+bv","3r)],[(1.9)=2xx a xx3" Riemann "(u","w","r)+2xx b xx3" Riemann "(u","v","r).]:}\begin{align*} & \boldsymbol{R i e m a n n}(2 \boldsymbol{u}, a \boldsymbol{w}+b \mathbf{v}, 3 \boldsymbol{r}) \\ = & 2 \times a \times 3 \text { Riemann }(\boldsymbol{u}, \boldsymbol{w}, \boldsymbol{r})+2 \times b \times 3 \text { Riemann }(\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{r}) . \tag{1.9} \end{align*}Riemann(2u,aw+bv,3r)(1.9)=2×a×3 Riemann (u,w,r)+2×b×3 Riemann (u,v,r).
Consequently, in any coordinate system the components of the vector put out can be written as a "trilinear function" of the components of the vectors put in:
(1.10) r = R i e m a n n ( u , v , w ) r α = R α β γ δ u β v γ w δ . (1.10) r = R i e m a n n ( u , v , w ) r α = R α β γ δ u β v γ w δ . {:(1.10)r=Riemann(u","v","w)<=>r^(alpha)=R^(alpha)_(beta gamma delta)u^(beta)v^(gamma)w^(delta).:}\begin{equation*} \boldsymbol{r}=\boldsymbol{\operatorname { R i e m a n n }}(\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}) \Leftrightarrow r^{\alpha}=R^{\alpha}{ }_{\beta \gamma \delta} u^{\beta} v^{\gamma} w^{\delta} . \tag{1.10} \end{equation*}(1.10)r=Riemann(u,v,w)rα=Rαβγδuβvγwδ.
(Here there is an implied summation on the indices β , γ , δ β , γ , δ beta,gamma,delta\beta, \gamma, \deltaβ,γ,δ; cf. Box 1.1.) The 4 × 4 × 4 × 4 = 256 4 × 4 × 4 × 4 = 256 4xx4xx4xx4=2564 \times 4 \times 4 \times 4=2564×4×4×4=256 numbers R α β γ δ R α β γ δ R^(alpha)_(beta gamma delta)R^{\alpha}{ }_{\beta \gamma \delta}Rαβγδ are called the "components of the Riemann tensor in the given coordinate system." In terms of components, the equation of geodesic deviation states
(1.8') D 2 ξ α d τ 2 + R α β γ δ d x β d τ ξ γ d x δ d τ = 0 . (1.8') D 2 ξ α d τ 2 + R α β γ δ d x β d τ ξ γ d x δ d τ = 0 . {:(1.8')(D^(2)xi^(alpha))/(dtau^(2))+R^(alpha)_(beta gamma delta)(dx^(beta))/(d tau)xi^(gamma)(dx^(delta))/(d tau)=0.:}\begin{equation*} \frac{D^{2} \xi^{\alpha}}{d \tau^{2}}+R^{\alpha}{ }_{\beta \gamma \delta} \frac{d x^{\beta}}{d \tau} \xi^{\gamma} \frac{d x^{\delta}}{d \tau}=0 . \tag{1.8'} \end{equation*}(1.8')D2ξαdτ2+Rαβγδdxβdτξγdxδdτ=0.
In Einstein's geometric theory of gravity, this equation of geodesic deviation summarizes the entire effect of geometry on matter. It does for gravitation physics what the Lorentz force equation,
(1.11) D 2 x α d τ 2 e m F β α d x α d τ = 0 (1.11) D 2 x α d τ 2 e m F β α d x α d τ = 0 {:(1.11)(D^(2)x^(alpha))/(dtau^(2))-(e)/(m)F_(beta)^(alpha)(dx^(alpha))/(d tau)=0:}\begin{equation*} \frac{D^{2} x^{\alpha}}{d \tau^{2}}-\frac{e}{m} F_{\beta}^{\alpha} \frac{d x^{\alpha}}{d \tau}=0 \tag{1.11} \end{equation*}(1.11)D2xαdτ2emFβαdxαdτ=0
does for electromagnetism. See Box 1.7.
The units of measurement of the curvature are cm 2 cm 2 cm^(-2)\mathrm{cm}^{-2}cm2 just as well in spacetime as on the surface of the apple. Nothing does so much to make these units stand out clearly as to express mass in "geometrized units":
Equation of geodesic deviation is analog of Lorentz force law
m ( cm ) = ( G / c 2 ) m conv ( g ) (1.12) = ( 0.742 × 10 28 cm / g ) m conv ( g ) m ( cm ) = G / c 2 m conv  ( g ) (1.12) = 0.742 × 10 28 cm / g m conv  ( g ) {:[m(cm)=(G//c^(2))m_("conv ")(g)],[(1.12)=(0.742 xx10^(-28)(cm)//g)m_("conv ")(g)]:}\begin{align*} m(\mathrm{~cm}) & =\left(G / c^{2}\right) m_{\text {conv }}(\mathrm{g}) \\ & =\left(0.742 \times 10^{-28} \mathrm{~cm} / \mathrm{g}\right) m_{\text {conv }}(\mathrm{g}) \tag{1.12} \end{align*}m( cm)=(G/c2)mconv (g)(1.12)=(0.742×1028 cm/g)mconv (g)
Box 1.7 EQUATION OF MOTION UNDER THE INFLUENCE OF A GRAVITATIONAL FIELD AND AN ELECTROMAGNETIC FIELD, COMPARED AND CONTRASTED
Electromagnetism
[Lorentz force, equation (1.11)]
Electromagnetism [Lorentz force, equation (1.11)]| Electromagnetism | | :--- | | [Lorentz force, equation (1.11)] |
Gravitation [Equation of geodesic deviation (1.8')]
Acceleration is defined for one particle? Yes No
Acceleration defined how? Actual world line compared to world line of uncharged "fiducial" test particle passing through same point with same 4 -velocity. Already an uncharged test particle, which can't accelerate relative to itself! Acceleration measured relative to a nearby test particle as fiduciary standard.
Acceleration depends on all four components of the 4 -velocity of the particle? Yes Yes
Universal acceleration for all test particles in same locations with same 4 -velocity? No ; is proportional to e / m e / m e//m\mathrm{e} / \mathrm{m}e/m Yes
Driving field Electromagnetic field Riemann curvature tensor
Ostensible number of distinct components of driving field 4 × 4 = 16 4 × 4 = 16 4xx4=164 \times 4=164×4=16 4 4 = 256 4 4 = 256 4^(4)=2564^{4}=25644=256
Actual number when allowance is made for symmetries of tensor 6 20
Names for more familiar of these components
3 electric
3 magnetic
3 electric 3 magnetic| 3 electric | | :--- | | 3 magnetic |
6 components of local Newtonian tide-producing acceleration
"Electromagnetism [Lorentz force, equation (1.11)]" Gravitation [Equation of geodesic deviation (1.8')] Acceleration is defined for one particle? Yes No Acceleration defined how? Actual world line compared to world line of uncharged "fiducial" test particle passing through same point with same 4 -velocity. Already an uncharged test particle, which can't accelerate relative to itself! Acceleration measured relative to a nearby test particle as fiduciary standard. Acceleration depends on all four components of the 4 -velocity of the particle? Yes Yes Universal acceleration for all test particles in same locations with same 4 -velocity? No ; is proportional to e//m Yes Driving field Electromagnetic field Riemann curvature tensor Ostensible number of distinct components of driving field 4xx4=16 4^(4)=256 Actual number when allowance is made for symmetries of tensor 6 20 Names for more familiar of these components "3 electric 3 magnetic" 6 components of local Newtonian tide-producing acceleration| | Electromagnetism <br> [Lorentz force, equation (1.11)] | Gravitation [Equation of geodesic deviation (1.8')] | | :---: | :---: | :---: | | Acceleration is defined for one particle? | Yes | No | | Acceleration defined how? | Actual world line compared to world line of uncharged "fiducial" test particle passing through same point with same 4 -velocity. | Already an uncharged test particle, which can't accelerate relative to itself! Acceleration measured relative to a nearby test particle as fiduciary standard. | | Acceleration depends on all four components of the 4 -velocity of the particle? | Yes | Yes | | Universal acceleration for all test particles in same locations with same 4 -velocity? | No ; is proportional to $\mathrm{e} / \mathrm{m}$ | Yes | | Driving field | Electromagnetic field | Riemann curvature tensor | | Ostensible number of distinct components of driving field | $4 \times 4=16$ | $4^{4}=256$ | | Actual number when allowance is made for symmetries of tensor | 6 | 20 | | Names for more familiar of these components | 3 electric <br> 3 magnetic | 6 components of local Newtonian tide-producing acceleration |
Components of Riemann tensor evaluated from relative accelerations of slowly moving particles
This conversion from grams to centimeters by means of the ratio
G / c 2 = 0.742 × 10 28 cm / g G / c 2 = 0.742 × 10 28 cm / g G//c^(2)=0.742 xx10^(-28)cm//gG / c^{2}=0.742 \times 10^{-28} \mathrm{~cm} / \mathrm{g}G/c2=0.742×1028 cm/g
is completely analogous to converting from seconds to centimeters by means of the ratio
c = 9.4605460000 × 10 17 cm 31 , 556 , 925.974 sec c = 9.4605460000 × 10 17 cm 31 , 556 , 925.974 sec c=(9.4605460000 xx10^(17)(cm))/(31,556,925.974sec)c=\frac{9.4605460000 \times 10^{17} \mathrm{~cm}}{31,556,925.974 \mathrm{sec}}c=9.4605460000×1017 cm31,556,925.974sec
(see end of §1.5). The sun, which in conventional units has m conv = 1.989 × 10 33 g m conv  = 1.989 × 10 33 g m_("conv ")=1.989 xx10^(33)gm_{\text {conv }}=1.989 \times 10^{33} \mathrm{~g}mconv =1.989×1033 g, has in geometrized units a mass m = 1.477 km m = 1.477 km m=1.477kmm=1.477 \mathrm{~km}m=1.477 km. Box 1.8 gives further discussion.
Using geometrized units, and using the Newtonian theory of gravity, one can readily evaluate nine of the most interesting components of the Riemann curvature tensor near the Earth or the sun. The method is the gravitational analog of determining the electric field strength by measuring the acceleration of a slowly moving test particle. Consider the separation between the geodesics of two nearby and slowly moving ( v c ) ( v c ) (v≪c)(v \ll c)(vc) particles at a distance r r rrr from the Earth or sun. In the standard, nearly inertial coordinates of celestial mechanics, all components of the 4 -velocity of the

Box 1.8 GEOMETRIZED UNITS

Throughout this book, we use "geometrized units," in which the speed of light c c ccc, Newton's gravitational constant G G GGG, and Boltzman's constant k k kkk are all equal to unity. The following alternative ways to express the number 1.0 are of great value:
1.0 = c = 2.997930 × 10 10 cm / sec 1.0 = G / c 2 = 0.7425 × 10 28 cm / g 1.0 = G / c 4 = 0.826 × 10 49 cm / erg 1.0 = G k / c 4 = 1.140 × 10 65 cm / K 1.0 = c 2 / G 1 / 2 = 3.48 × 10 24 cm / gauss 1 1.0 = c = 2.997930 × 10 10 cm / sec 1.0 = G / c 2 = 0.7425 × 10 28 cm / g 1.0 = G / c 4 = 0.826 × 10 49 cm / erg 1.0 = G k / c 4 = 1.140 × 10 65 cm / K 1.0 = c 2 / G 1 / 2 = 3.48 × 10 24 cm /  gauss  1 {:[1.0=c=2.997930 cdots xx10^(10)cm//sec],[1.0=G//c^(2)=0.7425 xx10^(-28)cm//g],[1.0=G//c^(4)=0.826 xx10^(-49)cm//erg],[1.0=Gk//c^(4)=1.140 xx10^(-65)cm//K],[1.0=c^(2)//G^(1//2)=3.48 xx10^(24)cm//" gauss "^(-1)]:}\begin{aligned} & 1.0=c=2.997930 \cdots \times 10^{10} \mathrm{~cm} / \mathrm{sec} \\ & 1.0=G / c^{2}=0.7425 \times 10^{-28} \mathrm{~cm} / \mathrm{g} \\ & 1.0=G / c^{4}=0.826 \times 10^{-49} \mathrm{~cm} / \mathrm{erg} \\ & 1.0=G k / c^{4}=1.140 \times 10^{-65} \mathrm{~cm} / \mathrm{K} \\ & 1.0=c^{2} / G^{1 / 2}=3.48 \times 10^{24} \mathrm{~cm} / \text { gauss }^{-1} \end{aligned}1.0=c=2.997930×1010 cm/sec1.0=G/c2=0.7425×1028 cm/g1.0=G/c4=0.826×1049 cm/erg1.0=Gk/c4=1.140×1065 cm/K1.0=c2/G1/2=3.48×1024 cm/ gauss 1
One can multiply a factor of unity, expressed in any one of these ways, into any term in any equation without affecting the validity of the equation. Thereby one can convert one's units of measure
from grams to centimeters to seconds to ergs to . . . . For example:
Mass of sun = M = 1.989 × 10 33 g = ( 1.989 × 10 33 g ) × ( G / c 2 ) = 1.477 × 10 5 cm = ( 1.989 × 10 33 g ) × ( c 2 ) = 1.788 × 10 54 ergs .  Mass of sun  = M = 1.989 × 10 33 g = 1.989 × 10 33 g × G / c 2 = 1.477 × 10 5 cm = 1.989 × 10 33 g × c 2 = 1.788 × 10 54 ergs . {:[" Mass of sun "=M_(o.)=1.989 xx10^(33)g],[=(1.989 xx10^(33)(g))xx(G//c^(2))],[=1.477 xx10^(5)cm],[=(1.989 xx10^(33)(g))xx(c^(2))],[=1.788 xx10^(54)ergs.]:}\begin{aligned} \text { Mass of sun } & =M_{\odot}=1.989 \times 10^{33} \mathrm{~g} \\ & =\left(1.989 \times 10^{33} \mathrm{~g}\right) \times\left(G / \mathrm{c}^{2}\right) \\ & =1.477 \times 10^{5} \mathrm{~cm} \\ & =\left(1.989 \times 10^{33} \mathrm{~g}\right) \times\left(c^{2}\right) \\ & =1.788 \times 10^{54} \mathrm{ergs} . \end{aligned} Mass of sun =M=1.989×1033 g=(1.989×1033 g)×(G/c2)=1.477×105 cm=(1.989×1033 g)×(c2)=1.788×1054ergs.
The standard unit, in terms of which everything is measured in this book, is centimeters. However, occasionally conventional units are used; in such cases a subscript "conv" is sometimes, but not always, appended to the quantity measured:
M c o n v = 1.989 × 10 33 g . M c o n v = 1.989 × 10 33 g . M_(o.conv)=1.989 xx10^(33)g.M_{\odot c o n v}=1.989 \times 10^{33} \mathrm{~g} .Mconv=1.989×1033 g.
fiducial test particle can be neglected except d x 0 / d τ = 1 d x 0 / d τ = 1 dx^(0)//d tau=1d x^{0} / d \tau=1dx0/dτ=1. The space components of the equation of geodesic deviation read
(1.13) d 2 ξ k / d τ 2 + R k 0 j 0 ξ j = 0 . (1.13) d 2 ξ k / d τ 2 + R k 0 j 0 ξ j = 0 . {:(1.13)d^(2)xi^(k)//dtau^(2)+R^(k)_(0j0)xi^(j)=0.:}\begin{equation*} d^{2} \xi^{k} / d \tau^{2}+R^{k}{ }_{0 j 0} \xi^{j}=0 . \tag{1.13} \end{equation*}(1.13)d2ξk/dτ2+Rk0j0ξj=0.
Comparing with the conclusions of Newtonian theory, equations (1.5), we arrive at the following information about the curvature of spacetime near a center of mass:
(units cm 2 cm 2 cm^(-2)\mathrm{cm}^{-2}cm2 ). Here and henceforth the caret or "hat" is used to indicate the components of a vector or tensor in a local Lorentz frame of reference ("physical components," as distinguished from components in a general coordinate system). Einstein's theory will determine the values of the other components of curvature (e.g., R x ^ z ^ x ^ z ^ = m / r 3 R x ^ z ^ x ^ z ^ = m / r 3 R^( hat(x)_( hat(z) hat(x) hat(z)))=-m//r^(3)R^{\hat{x}_{\hat{z} \hat{x} \hat{z}}}=-m / r^{3}Rx^z^x^z^=m/r3 ); but these nine terms are the ones of principal relevance for many applications of gravitation theory. They are analogous to the components of the electric field in the Lorentz equation of motion. Many of the terms not evaluated are analogous to magnetic field components-ordinarily weak unless the source is in rapid motion.
This ends the survey of the effect of geometry on matter ("effect of curvature of apple in causing geodesics to cross"-especially great near the dimple at the top, just as the curvature of spacetime is especially large near a center of gravitational attraction). Now for the effect of matter on geometry ("effect of stem of apple in causing dimple")!

§1.7. EFFECT OF MATTER ON GEOMETRY

The weight of any heavy body of known weight at a particular distance from the center of the world varies according to the variation of its distance therefrom; so that as often as it is removed from the center, it becomes heavier, and when brought near to it, is lighter. On this account, the relation of gravity to gravity is as the relation of distance to distance from the center.
AL KHĀZINī (Merv, A.D. 1115 ) ) ))), Book of the Balance of Wisdom
Figure 1.12 shows a sphere of the same density, ρ = 5.52 g / cm 3 ρ = 5.52 g / cm 3 rho=5.52g//cm^(3)\rho=5.52 \mathrm{~g} / \mathrm{cm}^{3}ρ=5.52 g/cm3, as the average density of the Earth. A hole is bored through this sphere. Two test particles, A A AAA and B B BBB, execute simple harmonic motion in this hole, with an 84 -minute period. Therefore their geodesic separation ξ ξ xi\xiξ, however it may be oriented, undergoes a simple periodic motion with the same 84 -minute period:
(1.15) d 2 ξ j / d τ 2 = ( 4 π 3 ρ ) ξ j , j = x or y or z . (1.15) d 2 ξ j / d τ 2 = 4 π 3 ρ ξ j , j = x  or  y  or  z . {:(1.15)d^(2)xi^(j)//dtau^(2)=-((4pi)/(3)rho)xi^(j)","quad j=x" or "y" or "z.:}\begin{equation*} d^{2} \xi^{j} / d \tau^{2}=-\left(\frac{4 \pi}{3} \rho\right) \xi^{j}, \quad j=x \text { or } y \text { or } z . \tag{1.15} \end{equation*}(1.15)d2ξj/dτ2=(4π3ρ)ξj,j=x or y or z.
Box 1.9 GALILEO GALILEI
Pisa, February 15, 1564-Arcetri, Florence, January 8, 1642

"In questions of science the authority of a thousand is not worth the humble reasoning of a single individual."
GALILEO GALILEI (1632)
"The spaces described by a body falling from rest with a uniformly accelerated motion are to each other as the squares of the time intervals employed in traversing these distances."
GALILEO GALILEI (1638)
"Everything that has been said before and imagined by other people [about the tides] is in my opinion completely invalid. But among the great men who have philosophised about this marvellous effect of nature the one who surprised me the most is Kepler. More than other people he was a person of independent genius, sharp, and had in his hands the motion of the earth. He later pricked up his ears and became interested in the action of the moon on the water, and in other occult phenomena, and similar childishness."
GALILEO GALILEI (1632)
"It is a most beautiful and delightful sight to behold [with the new telescope] the body of the Moon . . . the Moon certainly does not possess a smooth and polished surface, but one rough and uneven . . . full of vast protuberances, deep chasms and sinuosities . . . stars in myriads, which have never been seen before and which surpass the old, previously known, stars in number more than ten times. I have discovered four planets, neither known nor observed by any one of the astronomers before my time . . . got rid of disputes about the Galaxy or Milky Way, and made its nature clear to the very senses, not to say to the understanding . . . the galaxy is nothing else than a mass of luminous stars planted together in clusters . . . the number of small ones is quite beyond determination - the stars which have been called by every one of the astronomers up to this day nebulous are groups of small stars set thick together in a wonderful
way. "
GALILEO GALILEI IN SIDEREUS NUNCIUS (1610)
"So the principles which are set forth in this treatise will, when taken up by thoughtful minds, lead to many another more remarkable result; and it is to be believed that it will be so on account of the nobility of the subject, which is superior to any other in nature."
Figure 1.12.
Test particles A A AAA and B B BBB move up and down a hole bored through the Earth, idealized as of uniform density. At radius r r rrr, a particle feels Newtonian acceleration
d 2 r d τ 2 = 1 c 2 d 2 r d t conv 2 = G c 2 (mass inside radius r ) r 2 = ( G r 2 c 2 ) ( 4 π 3 ρ conv r 3 ) = ω 2 r . d 2 r d τ 2 = 1 c 2 d 2 r d t conv  2 = G c 2  (mass inside radius  r  )  r 2 = G r 2 c 2 4 π 3 ρ conv  r 3 = ω 2 r . {:[(d^(2)r)/(dtau^(2))=(1)/(c^(2))(d^(2)r)/(dt_("conv ")^(2))],[=-(G)/(c^(2))(" (mass inside radius "r" ) ")/(r^(2))],[=-((G)/(r^(2)c^(2)))((4pi)/(3)rho_("conv ")r^(3))],[=-omega^(2)r.]:}\begin{aligned} \frac{d^{2} r}{d \tau^{2}} & =\frac{1}{c^{2}} \frac{d^{2} r}{d t_{\text {conv }}{ }^{2}} \\ & =-\frac{G}{c^{2}} \frac{\text { (mass inside radius } r \text { ) }}{r^{2}} \\ & =-\left(\frac{G}{r^{2} c^{2}}\right)\left(\frac{4 \pi}{3} \rho_{\text {conv }} r^{3}\right) \\ & =-\omega^{2} r . \end{aligned}d2rdτ2=1c2d2rdtconv 2=Gc2 (mass inside radius r ) r2=(Gr2c2)(4π3ρconv r3)=ω2r.
Consequently, each particle oscillates in simple harmonic motion with precisely the same angular frequency as a satellite, grazing the model Earth, traverses its circular orbit:
ω 2 ( cm 2 ) = 4 π 3 ρ ( cm 2 ) ω conv 2 ( sec 2 ) = 4 π G 3 ρ conv ( g / cm 3 ) ω 2 cm 2 = 4 π 3 ρ cm 2 ω conv  2 sec 2 = 4 π G 3 ρ conv  g / cm 3 {:[omega^(2)(cm^(-2))=(4pi)/(3)rho(cm^(-2))],[omega_("conv ")^(2)(sec^(-2))=(4pi G)/(3)rho_("conv ")(g//cm^(3))]:}\begin{aligned} \omega^{2}\left(\mathrm{~cm}^{-2}\right) & =\frac{4 \pi}{3} \rho\left(\mathrm{~cm}^{-2}\right) \\ \omega_{\text {conv }}^{2}\left(\mathrm{sec}^{-2}\right) & =\frac{4 \pi G}{3} \rho_{\text {conv }}\left(\mathrm{g} / \mathrm{cm}^{3}\right) \end{aligned}ω2( cm2)=4π3ρ( cm2)ωconv 2(sec2)=4πG3ρconv (g/cm3)
Comparing this actual motion with the equation of geodesic deviation (1.13) for slowly moving particles in a nearly inertial frame, we can read off some of the curvature components for the interior of this model Earth.
This example illustrates how the curvature of spacetime is connected to the distribution of matter.
Let a gravitational wave from a supernova pass through the Earth. Idealize the Earth's matter as so nearly incompressible that its density remains practically unchanged. The wave is characterized by ripples in the curvature of spacetime, propagating with the speed of light. The ripples will show up in the components R j 0 k 0 R j 0 k 0 R^(j)_(0k0)R^{j}{ }_{0 k 0}Rj0k0 of the Riemann tensor, and in the relative acceleration of our two test particles. The left side of equation (1.16) will ripple; but the right side will not. Equation (1.16) will break down. No longer will the Riemann curvature be generated directly and solely by the Earth's matter.
Nevertheless, Einstein tells us, a part of equation (1.16) is undisturbed by the
The Riemann tensor inside the Earth
Effect of gravitational wave on Riemann tensor

waves: its trace
(1.17) R 0 ^ 0 ^ R x ^ θ ^ x ^ 0 ^ 0 ^ + R y ^ 0 ^ y ^ 0 ^ ^ + R z ^ 0 ^ z ^ 0 ^ ^ = 4 π ρ . (1.17) R 0 ^ 0 ^ R x ^ θ ^ x ^ 0 ^ 0 ^ + R y ^ 0 ^ y ^ 0 ^ ^ + R z ^ 0 ^ z ^ 0 ^ ^ = 4 π ρ . {:(1.17)R_( hat(0) hat(0))-=R^( hat(x)) hat(theta)_( hat(x) hat(0) hat(0))+R^( hat(y)) hat(( hat(0))( hat(y))( hat(0)))+R^( hat(z)) hat(( hat(0))( hat(z))( hat(0)))=4pi rho.:}\begin{equation*} R_{\hat{0} \hat{0}} \equiv R^{\hat{x}} \hat{\theta}_{\hat{x} \hat{0} \hat{0}}+R^{\hat{y}} \hat{\hat{0} \hat{y} \hat{0}}+R^{\hat{z}} \hat{\hat{0} \hat{z} \hat{0}}=4 \pi \rho . \tag{1.17} \end{equation*}(1.17)R0^0^Rx^θ^x^0^0^+Ry^0^y^0^^+Rz^0^z^0^^=4πρ.
Even in the vacuum outside the Earth this is valid; there both sides vanish [cf. (1.14)].
More generally, a certain piece of the Riemann tensor, called the Einstein tensor and denoted Einstein or G G G\boldsymbol{G}G, is always generated directly by the local distribution of matter. Einstein is the geometric object that generalizes R 0 ^ 0 ^ R 0 ^ 0 ^ R_( hat(0) hat(0))R_{\hat{0} \hat{0}}R0^0^, the lefthand side
Einstein tensor introduced
"The description of right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn."
[FROM P. 1 OF NEWTON'S PREFACE TO THE FIRST (1687) EDITION OF THE PRINCIPIA]
"Absolute space, in its own nature, without relation to anything external, remains always similar and immovable
"Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external."
[FROM THE SCHOLIUM IN THE PRINCIPIA]
"I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypotheses; for whatever is not reduced from the phenomena is to be called an hypothesis; and hypotheses . . . have no place in experimental philosophy. . . And to us it is enough that gravity does really exist, and act according to the laws which we have explained, and abundantly serves to account for all the motions of the celestial bodies, and of our sea."
[FROM THE GENERAL SCHOLIUM ADDED AT THE END OF THE THIRD BOOK OF THE PRINCIPIA IN THE SECOND EDITION OF 1713; ESPECIALLY FAMOUS FOR THE PHRASE OFTEN QUOTED FROM NEWTON'S ORIGINAL LATIN, "HYPOTHESES NON FINGO.'"]
"And the same year [1665 or 1666] I began to think of gravity extending to the orb of the Moon, and having found out. ... All this was in the two plague years of 1665 and 1666 , for in those days I was in the prime of my age for invention, and minded Mathematicks and Philosophy more than at any time since."
[FROM MEMORANDUM IN NEWTON'S HANDWRITING ABOUT HIS DISCOVERIES ON FLUXIONS, THE BINOMIL THEOREM. OPTICS. DYNAMICS. AND GRAVITY, BEIEVED TO HAVE BEEN WRITTEN ABOUT 1714. AND FOUND BY ADAMS ABOUT 1887 IN THE "PORTSMOUTH COLLECTION" OF
NEWTON PAPERS]
of equation (1.17). Like R 0 ^ 0 ^ R 0 ^ 0 ^ R_( hat(0) hat(0))R_{\hat{0} \hat{0}}R0^0^, Einstein is a sort of average of Riemann over all directions. Generating Einstein and generalizing the righthand side of (1.16) is a geometric object called the stress-energy tensor of the matter. It is denoted T T T\boldsymbol{T}T. No coordinates are need to define Einstein, and none to define T T T\boldsymbol{T}T; like the Riemann tensor, Riemann, and the metric tensor, g g g\boldsymbol{g}g, they exist in the complete absence of coordinates. Moreover, in nature they are always equal, aside from a factor of 8 π 8 π 8pi8 \pi8π :
(1.18) Einstein G = 8 π T (1.18)  Einstein  G = 8 π T {:(1.18)" Einstein "-=G=8pi T:}\begin{equation*} \text { Einstein } \equiv \boldsymbol{G}=8 \pi \boldsymbol{T} \tag{1.18} \end{equation*}(1.18) Einstein G=8πT
Stress-energy tensor introduced
"For hypotheses ought . . . to explain the properties of things and not attempt to predetermine them except in so far as they can be an aid to experiments." [FROM LETTER OF NEWTON TO I. M. PARDIES, 1672, AS QUoted IN THE CAJORI NOTES AT THE END OF NEWTON (1687), P. 673]
"That one body may act upon another at a distance through a vacuum, without the mediation of any thing else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity, that I believe no man, who has in philosophical matters a competent faculty of thinking, can ever fall into it."
[PASSAGE OFTEN QUOTED BY MICHAEL FARADAY FROM LETTERS OF NEWTON TO RICHARD BENTLY, 1692-1693, AS QUOTED IN THE NOTES OF THE CAJORI EDITION OF NEWTON (1687), P.
643]
"The attractions of gravity, magnetism, and electricity, reach to very sensible distances, and so have been observed . . ; and there may be others which reach
to so small distances as hitherto escape observation; . . . some force, which in immediate contract is exceeding strong, at small distances performs the chemical operations above-mentioned, and reaches not far from the particles with any sensible effect."
[from query 31 at the end of newton's opticks (1730)]
"What is there in places almost empty of matter, and whence is it that the sun and planets gravitate towards one another, without dense matter between them? Whence is it that nature doth nothing in vain; and whence arises all that order and beauty which we see in the world? To what end are comets, and whence is it that planets move all one and the same way in orbs concentrick, while comets move all manner of ways in orbs very excentrick; and what hinders the fixed stars from falling upon one another?"
[FROM QUERY 28]
Consequences of Einstein field equation
This Einstein field equation, rewritten in terms of components in an arbitrary coordinate system, reads
(1.19) G α β = 8 π T α β (1.19) G α β = 8 π T α β {:(1.19)G_(alpha beta)=8piT_(alpha beta):}\begin{equation*} G_{\alpha \beta}=8 \pi T_{\alpha \beta} \tag{1.19} \end{equation*}(1.19)Gαβ=8πTαβ
The Einstein field equation is elegant and rich. No equation of physics can be written more simply. And none contains such a treasure of applications and consequences.
The field equation shows how the stress-energy of matter generates an average curvature (Einstein G ) G ) -=G)\equiv \boldsymbol{G})G) in its neighborhood. Simultaneously, the field equation is a propagation equation for the remaining, anisotropic part of the curvature: it governs the external spacetime curvature of a static source (Earth); it governs the generation of gravitational waves (ripples in curvature of spacetime) by stress-energy in motion; and it governs the propagation of those waves through the universe. The field equation even contains within itself the equations of motion ("Force = = ===

mass × × xx\times× acceleration") for the matter whose stress-energy generates the curvature.
Those were some consequences of G = 8 π T G = 8 π T G=8pi T\boldsymbol{G}=8 \pi \boldsymbol{T}G=8πT. Now for some applications.
The field equation governs the motion of the planets in the solar system; it governs the deflection of light by the sun; it governs the collapse of a star to form a black hole; it determines uniquely the external spacetime geometry of a black hole ("a black hole has no hair"); it governs the evolution of spacetime singularities at the end point of collapse; it governs the expansion and recontraction of the universe. And more; much more.
In order to understand how the simple equation G = 8 π T G = 8 π T G=8pi T\boldsymbol{G}=8 \pi \boldsymbol{T}G=8πT can be so all powerful, it is desirable to backtrack, and spend a few chapters rebuilding the entire picture of spacetime, of its curvature, and of its laws, this time with greater care, detail, and mathematics.
Thus ends this survey of the effect of geometry on matter, and the reaction of matter back on geometry, rounding out the parable of the apple.
Applications of Einstein field equation
"What really interests me is whether God had any choice in the creation of the world"
EINSTEIN TO AN ASSISTANT, AS QUOTED BY G. HOLTON (1971), P. 20
"But the years of anxious searching in the dark, with their intense longing, their alternations of confidence and exhaustion, and the final emergence into the light-only those who have experienced it can understand that" EINSTEIN, AS QUOTED BY M. KLEIN (1971), P. 1315
"Of all the communities available to us there is not one I would want to devote myself to, except for the society of the true searchers, which has very few living members at any time. . ."
EINSTEIN LETTER TO BORN, QUOTED BY BORN (1971), P. 82
"I am studying your great works and -when I get stuck anywhere-now have the pleasure of seeing your friendly young face before me smiling and explaining" EINSTEIN, LETTER OF MAY 2, 1920, AFTER MEETING NIELS bOhr
"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." EINSTEIN (1921), P. 28
"The most incomprehensible thing about the world is that it is comprehensible." EINSTEIN, IN SCHILPP (1949), P. 112

EXERCISES

Exercise 1.1. CURVATURE OF A CYLINDER

Show that the Gaussian curvature R R RRR of the surface of a cylinder is zero by showing that geodesics on that surface (unroll!) suffer no geodesic deviation. Give an independent argument for the same conclusion by employing the formula R = 1 / ρ 1 ρ 2 R = 1 / ρ 1 ρ 2 R=1//rho_(1)rho_(2)R=1 / \rho_{1} \rho_{2}R=1/ρ1ρ2, where ρ 1 ρ 1 rho_(1)\rho_{1}ρ1 and ρ 2 ρ 2 rho_(2)\rho_{2}ρ2 are the principal radii of curvature at the point in question with respect to the enveloping Euclidean three-dimensional space.

Exercise 1.2. SPRING TIDE VS. NEAP TIDE

Evaluate (1) in conventional units and (2) in geometrized units the magnitude of the Newtonian tide-producing acceleration R m 0 n 0 ( m , n = 1 , 2 , 3 ) R m 0 n 0 ( m , n = 1 , 2 , 3 ) R^(m)_(0n0)(m,n=1,2,3)R^{m}{ }_{0 n 0}(m, n=1,2,3)Rm0n0(m,n=1,2,3) generated at the Earth by (1) the moon ( m conv = 7.35 × 10 25 g , r = 3.84 × 10 10 cm m conv  = 7.35 × 10 25 g , r = 3.84 × 10 10 cm m_("conv ")=7.35 xx10^(25)g,r=3.84 xx10^(10)cmm_{\text {conv }}=7.35 \times 10^{25} \mathrm{~g}, r=3.84 \times 10^{10} \mathrm{~cm}mconv =7.35×1025 g,r=3.84×1010 cm ) and (2) the sun ( m conv = 1.989 × 10 33 g m conv  = 1.989 × 10 33 g m_("conv ")=1.989 xx10^(33)gm_{\text {conv }}=1.989 \times 10^{33} \mathrm{~g}mconv =1.989×1033 g, r = 1.496 × 10 13 cm ) r = 1.496 × 10 13 cm {:r=1.496 xx10^(13)(cm))\left.r=1.496 \times 10^{13} \mathrm{~cm}\right)r=1.496×1013 cm). By what factor do you expect spring tides to exceed neap tides?

Exercise 1.3. KEPLER ENCAPSULATED

A small satellite has a circular frequency ω ( cm 1 ) ω cm 1 omega(cm^(-1))\omega\left(\mathrm{cm}^{-1}\right)ω(cm1) in an orbit of radius r r rrr about a central object of mass m ( cm ) m ( cm ) m(cm)m(\mathrm{~cm})m( cm). From the known value of ω ω omega\omegaω, show that it is possible to determine neither r r rrr nor m m mmm individually, but only the effective "Kepler density" of the object as averaged over a sphere of the same radius as the orbit. Give the formula for ω 2 ω 2 omega^(2)\omega^{2}ω2 in terms of this Kepler density.
It is a reminder of the continuity of history that Kepler and Galileo (Box 1.9) wrote back and forth, and that the year that witnessed the death of Galileo saw the birth of Newton (Box 1.10). After Newton the first dramatically new synthesis of the laws of gravitation came from Einstein (Box 1.11).
And what the dead had no speech for, when living,
They can tell you, being dead; the communication Of the dead is tongued with fire beyond the language of the living.
T. S. ELIOT, in LITTLE GIDDING (1942)
I measured the skies Now the shadows I measure
Skybound was the mind Earthbound the body rests

PHYSICS IN FLAT SPACETIME

Wherein the reader meets an old friend, Special Relativity, outfitted in new, mod attire, and becomes more intimately acquainted with her charms.

FOUNDATIONS OF SPECIAL RELATIVITY

In geometric and physical applications, it always turns out that a quantity is characterized not only by its tensor order, but also by symmetry.HERMAN WEYL (1925)

Undoubtedly the most striking development of geometry during the last 2,000 years is the continual expansion of the concept "geometric object." This concept began by comprising only the few curves and surfaces of Greek synthetic geometry; it was stretched, during the Renaissance, to cover the whole domain of those objects defined by analytic geometry; more recently, it has been extended to cover the boundless universe treated by point-set theory.
KARL MENGER, IN SCHILPP (1949), P. 466.

§2.1. OVERVIEW

Curvature in geometry manifests itself as gravitation. Gravitation works on the separation of nearby particle world lines. In turn, particles and other sources of mass-energy cause curvature in the geometry. How does one break into this closed loop of the action of geometry on matter and the reaction of matter on geometry? One can begin no better than by analyzing the motion of particles and the dynamics of fields in a region of spacetime so limited that it can be regarded as flat. (See "Test for Flatness," Box 1.5).
Chapters 2-6 develop this flat-spacetime viewpoint (special relativity). The reader, it is assumed, is already somewhat familiar with special relativity:* 4 -vectors in
Background assumed of reader general; the energy-momentum 4-vector; elementary Lorentz transformations; the Lorentz law for the force on a charged particle; at least one look at one equation
in one book that refers to the electromagnetic field tensor F μ ν F μ ν F_(mu nu)F_{\mu \nu}Fμν; and the qualitative features of spacetime diagrams, including such points as (1) future and past light cones, (2) causal relationships ("past of," "future of," "neutral," or "in a spacelike relationship to"), (3) Lorentz contraction, (4) time dilation, (5) absence of a universal concept of simultaneity, and (6) the fact that the T ¯ T ¯ bar(T)\bar{T}T¯ and z ¯ z ¯ bar(z)\bar{z}z¯ axes in Box 2.4 are orthogonal even though they do not look so. If the reader finds anything new in these chapters, it will be: (i) a new viewpoint on special relativity, one emphasizing coordinate-free concepts and notation that generalize readily to curved spacetime ("geometric objects," tensors viewed as machines-treated in Chapters 2-4); or (ii) unfamiliar topics in special relativity, topics crucial to the later exposition of gravitation theory ("stress-energy tensor and conservation laws," Chapter 5; "accelerated observers," Chapter 6).

§2.2. GEOMETRIC OBJECTS

Every physical quantity can be described by a geometric object
All laws of physics can be expressed geometrically
Everything that goes on in spacetime has its geometric description, and almost every one of these descriptions lends itself to ready generalization from flat spacetime to curved spacetime. The greatest of the differences between one geometric object and another is its scope: the individual object (vector) for the momentum of a certain particle at a certain phase in its history, as contrasted to the extended geometric object that describes an electromagnetic field defined throughout space and time ("antisymmetric second-rank tensor field" or, more briefly, "field of 2-forms"). The idea that every physical quantity must be describable by a geometric object, and that the laws of physics must all be expressible as geometric relationships between these geometric objects, had its intellectual beginnings in the Erlanger program of Felix Klein (1872), came closer to physics in Einstein's "principle of general covariance" and in the writings of Hermann Weyl (1925), seems to have first been formulated clearly by Veblen and Whitehead (1932), and today pervades relativity theory, both special and general.
A. Nijenhuis (1952) and S.-S. Chern ( 1960 , 1966 , 1971 ) ( 1960 , 1966 , 1971 ) (1960,1966,1971)(1960,1966,1971)(1960,1966,1971) have expounded the mathematical theory of geometric objects. But to understand or do research in geometrodynamics, one need not master this elegant and beautiful subject. One need only know that geometric objects in spacetime are entities that exist independently of coordinate systems or reference frames. A point in spacetime ("event") is a geometric object. The arrow linking two neighboring events ("vector") is a geometric object in flat spacetime, and its generalization, the "tangent vector," is a geometric object even when spacetime is curved. The "metric" (machine for producing the squared length of any vector; see Box 1.3) is a geometric object. No coordinates are needed to define any of these concepts.
The next few sections will introduce several geometric objects, and show the roles they play as representatives of physical quantities in flat spacetime.
Figure 2.1.
From vector as connector of two points to vector as derivative ("tangent vector"; a local rather than a bilocal concept).

§2.3. VECTORS

Begin with the simplest idea of a vector (Figure 2.1B): an arrow extending from one spacetime event A A A\mathscr{A}A ("tail") to another event B B B\mathscr{B}B ("tip"). Write this vector as
v a B = B a ( or A B ) . v a B = B a (  or  A B ) . v_(aB)=B-a(" or "AB).\boldsymbol{v}_{a B}=\mathscr{B}-a(\text { or } \mathscr{A} \mathscr{B}) .vaB=Ba( or AB).
For many purposes (including later generalization to curved spacetime) other completely equivalent ways to think of this vector are more convenient. Represent the arrow by the parametrized straight line P ( λ ) = a + λ ( B C ) P ( λ ) = a + λ ( B C ) P(lambda)=a+lambda(B-C)\mathscr{P}(\lambda)=\mathscr{a}+\lambda(\mathscr{B}-\mathscr{C})P(λ)=a+λ(BC), with λ = 0 λ = 0 lambda=0\lambda=0λ=0 the tail of the arrow, and λ = 1 λ = 1 lambda=1\lambda=1λ=1 its tip. Form the derivative of this simple linear expression for P ( λ ) P ( λ ) P(lambda)\mathscr{P}(\lambda)P(λ) :
( d / d λ ) [ a + λ ( B a ) ] = B a = P ( 1 ) P ( 0 ) ( tip ) ( tail ) v G B . ( d / d λ ) [ a + λ ( B a ) ] = B a = P ( 1 ) P ( 0 ) (  tip  ) (  tail  ) v G B . (d//d lambda)[a+lambda(B-a)]=B-a=P(1)-P(0)-=(" tip ")-(" tail ")-=v_(GB).(d / d \lambda)[a+\lambda(\mathscr{B}-\mathscr{a})]=\mathscr{B}-a=\mathscr{P}(1)-\mathscr{P}(0) \equiv(\text { tip })-(\text { tail }) \equiv \boldsymbol{v}_{G \mathscr{B}} .(d/dλ)[a+λ(Ba)]=Ba=P(1)P(0)( tip )( tail )vGB.
This result allows one to replace the idea of a vector as a 2-point object ("bilocal") by the concept of a vector as a 1-point object ("tangent vector"; local):
(2.1) v G B = ( d P / d λ ) λ = 0 . (2.1) v G B = ( d P / d λ ) λ = 0 . {:(2.1)v_(GB)=(dP//d lambda)_(lambda=0).:}\begin{equation*} \boldsymbol{v}_{G \mathscr{B}}=(d \mathscr{P} / d \lambda)_{\lambda=0} . \tag{2.1} \end{equation*}(2.1)vGB=(dP/dλ)λ=0.
Example: if P ( τ ) P ( τ ) P(tau)\mathscr{P}(\tau)P(τ) is the straight world line of a free particle, parametrized by its proper time, then the displacement that occurs in a proper time interval of one second gives an arrow u = P ( 1 ) P ( 0 ) u = P ( 1 ) P ( 0 ) u=P(1)-P(0)\boldsymbol{u}=\mathscr{P}(1)-\mathscr{P}(0)u=P(1)P(0). This arrow is easily drawn on a spacetime diagram. It accurately shows the 4 -velocity of the particle. However, the derivative formula u = d P / d τ u = d P / d τ u=dP//d tau\boldsymbol{u}=d \mathscr{P} / d \tauu=dP/dτ for computing the same displacement (1) is more suggestive of the velocity concept and (2) lends itself to the case of accelerated motion. Thus, given a world line P ( τ ) P ( τ ) P(tau)\mathscr{P}(\tau)P(τ) that is not straight, as in Figure 2.2, one must first form d P / d τ d P / d τ dP//d taud \mathscr{P} / d \taudP/dτ, and only thereafter draw the straight line P ( 0 ) + λ ( d P / d τ ) 0 P ( 0 ) + λ ( d P / d τ ) 0 P(0)+lambda(dP//d tau)_(0)\mathscr{P}(0)+\lambda(d \mathscr{P} / d \tau)_{0}P(0)+λ(dP/dτ)0 of the arrow u = d P / d τ u = d P / d τ u=dP//d tau\boldsymbol{u}=d \mathscr{P} / d \tauu=dP/dτ to display the 4 -velocity u u u\boldsymbol{u}u.
Ways of defining vector: As arrow
As parametrized straight line
As derivative of point along curve
Figure 2.2.
Same tangent vector derived from two very different curves. That parametrized straight line is also drawn which best fits the two curves at P 0 P 0 P_(0)\mathscr{\mathscr { P }}_{0}P0. The tangent vector reaches from 0 to 1 on this straight line.
Basis vectors
The reader may be unfamiliar with this viewpoint. More familiar may be the components of the 4 -velocity in a specific Lorentz reference frame:
(2.2) u 0 = d t d τ = 1 1 v 2 , u j = d x j d τ = v j 1 v 2 (2.2) u 0 = d t d τ = 1 1 v 2 , u j = d x j d τ = v j 1 v 2 {:(2.2)u^(0)=(dt)/(d tau)=(1)/(sqrt(1-v^(2)))","quadu^(j)=(dx^(j))/(d tau)=(v^(j))/(sqrt(1-v^(2))):}\begin{equation*} u^{0}=\frac{d t}{d \tau}=\frac{1}{\sqrt{1-v^{2}}}, \quad u^{j}=\frac{d x^{j}}{d \tau}=\frac{v^{j}}{\sqrt{1-v^{2}}} \tag{2.2} \end{equation*}(2.2)u0=dtdτ=11v2,uj=dxjdτ=vj1v2
where
v j = d x j / d t = components of "ordinary velocity," v 2 = ( v x ) 2 + ( v y ) 2 + ( v z ) 2 . v j = d x j / d t =  components of "ordinary velocity,"  v 2 = v x 2 + v y 2 + v z 2 . {:[v^(j)=dx^(j)//dt=" components of "ordinary velocity," "],[v^(2)=(v^(x))^(2)+(v^(y))^(2)+(v^(z))^(2).]:}\begin{aligned} & v^{j}=d x^{j} / d t=\text { components of "ordinary velocity," } \\ & v^{2}=\left(v^{x}\right)^{2}+\left(v^{y}\right)^{2}+\left(v^{z}\right)^{2} . \end{aligned}vj=dxj/dt= components of "ordinary velocity," v2=(vx)2+(vy)2+(vz)2.
Even the components (2.2) of 4 -velocity may seem slightly unfamiliar if the reader is accustomed to having the fourth component of a vector be multiplied by a factor i = 1 i = 1 i=sqrt(-1)i=\sqrt{-1}i=1. If so, he must adjust himself to new notation. (See "Farewell to 'ict,'" Box 2.1.)
More fundamental than the components of a vector is the vector itself. It is a geometric object with a meaning independent of all coordinates. Thus a particle has a world line P ( τ ) P ( τ ) P(tau)\mathscr{P}(\tau)P(τ), and a 4-velocity u = d P / d τ u = d P / d τ u=dP//d tau\boldsymbol{u}=d \mathscr{P} / d \tauu=dP/dτ, that have nothing to do with any coordinates. Coordinates enter the picture when analysis on a computer is required (rejects vectors; accepts numbers). For this purpose one adopts a Lorentz frame with orthonormal basis vectors (Figure 2.3) e 0 , e 1 , e 2 e 0 , e 1 , e 2 e_(0),e_(1),e_(2)\boldsymbol{e}_{0}, \boldsymbol{e}_{1}, \boldsymbol{e}_{2}e0,e1,e2, and e 3 e 3 e_(3)\boldsymbol{e}_{3}e3. Relative to the origin O O O\mathcal{O}O of this frame, the world line has a coordinate description
P ( τ ) O = x 0 ( τ ) e 0 + x 1 ( τ ) e 1 + x 2 ( τ ) e 2 + x 3 ( τ ) e 3 = x μ ( τ ) e μ . P ( τ ) O = x 0 ( τ ) e 0 + x 1 ( τ ) e 1 + x 2 ( τ ) e 2 + x 3 ( τ ) e 3 = x μ ( τ ) e μ . P(tau)-O=x^(0)(tau)e_(0)+x^(1)(tau)e_(1)+x^(2)(tau)e_(2)+x^(3)(tau)e_(3)=x^(mu)(tau)e_(mu).\mathscr{P}(\tau)-\mathcal{O}=x^{0}(\tau) \boldsymbol{e}_{0}+x^{1}(\tau) \boldsymbol{e}_{1}+x^{2}(\tau) \boldsymbol{e}_{2}+x^{3}(\tau) \boldsymbol{e}_{3}=x^{\mu}(\tau) \boldsymbol{e}_{\mu} .P(τ)O=x0(τ)e0+x1(τ)e1+x2(τ)e2+x3(τ)e3=xμ(τ)eμ.
Expressed relative to the same Lorentz frame, the 4 -velocity of the particle is
(2.3) u = d P / d τ = ( d x μ / d τ ) e μ = u 0 e 0 + u 1 e 1 + u 2 e 2 + u 3 e 3 (2.3) u = d P / d τ = d x μ / d τ e μ = u 0 e 0 + u 1 e 1 + u 2 e 2 + u 3 e 3 {:(2.3)u=dP//d tau=(dx^(mu)//d tau)e_(mu)=u^(0)e_(0)+u^(1)e_(1)+u^(2)e_(2)+u^(3)e_(3):}\begin{equation*} \boldsymbol{u}=d \mathscr{P} / d \tau=\left(d x^{\mu} / d \tau\right) \boldsymbol{e}_{\mu}=u^{0} \boldsymbol{e}_{0}+u^{1} \boldsymbol{e}_{1}+u^{2} \boldsymbol{e}_{2}+u^{3} \boldsymbol{e}_{3} \tag{2.3} \end{equation*}(2.3)u=dP/dτ=(dxμ/dτ)eμ=u0e0+u1e1+u2e2+u3e3

Box 2.1 FAREWELL TO "ict"

One sometime participant in special relativity will have to be put to the sword: " x 4 = i c t x 4 = i c t x^(4)=ictx^{4}=i c tx4=ict." This imaginary coordinate was invented to make the geometry of spacetime look formally as little different as possible from the geometry of Euclidean space; to make a Lorentz transformation look on paper like a rotation; and to spare one the distinction that one otherwise is forced to make between quantities with upper indices (such as the components p μ p μ p^(mu)p^{\mu}pμ of the energy-momentum vector) and quantities with lower indices (such as the components p μ p μ p_(mu)p_{\mu}pμ of the energy-momentum 1-form). However, it is no kindness to be spared this latter distinction. Without it, one cannot know whether a vector ( $ 2.3 ) ( $ 2.3 ) ($2.3)(\$ 2.3)($2.3) is meant or the very different geometric object that is a 1 -form ( $ 2.5 $ 2.5 $2.5\$ 2.5$2.5 ). Moreover, there is a significant difference between an angle on which everything depends periodically (a rotation) and a parameter the increase of which gives rise to ever-growing momentum differences (the "velocity parameter" of a Lorentz transformation; Box 2.4). If the imaginary time-coordinate hides from view the character of the geometric object being dealt with and the nature of the parameter in a transformation, it also does something even more serious: it hides the completely different metric structure ( § 2.4 ) ( § 2.4 ) (§2.4)(\S 2.4)(§2.4) of +++ geometry and -+++ geometry. In Euclidean geometry, when the distance between two points is zero, the two
points must be the same point. In Lorentz-Minkowski geometry, when the interval between two events is zero, one event may be on Earth and the other on a supernova in the galaxy M31, but their separation must be a null ray (piece of a light cone). The backward-pointing light cone at a given event contains all the events by which that event can be influenced. The forward-pointing light cone contains all events that it can influence. The multitude of double light cones taking off from all the events of spacetime forms an interlocking causal structure. This structure makes the machinery of the physical world function as it does (further comments on this structure in Wheeler and Feynman 1945 and 1949 and in Zeeman 1964). If in a region where spacetime is flat, one can hide this structure from view by writing
( Δ s ) 2 = ( Δ x 1 ) 2 + ( Δ x 2 ) 2 + ( Δ x 3 ) 2 + ( Δ x 4 ) 2 ( Δ s ) 2 = Δ x 1 2 + Δ x 2 2 + Δ x 3 2 + Δ x 4 2 (Delta s)^(2)=(Deltax^(1))^(2)+(Deltax^(2))^(2)+(Deltax^(3))^(2)+(Deltax^(4))^(2)(\Delta s)^{2}=\left(\Delta x^{1}\right)^{2}+\left(\Delta x^{2}\right)^{2}+\left(\Delta x^{3}\right)^{2}+\left(\Delta x^{4}\right)^{2}(Δs)2=(Δx1)2+(Δx2)2+(Δx3)2+(Δx4)2
with x 4 = i c t x 4 = i c t x^(4)=ictx^{4}=i c tx4=ict, no one has discovered a way to make an imaginary coordinate work in the general curved spacetime manifold. If " x 4 = i c t x 4 = i c t x^(4)=ictx^{4}=i c tx4=ict " cannot be used there, it will not be used here. In this chapter and hereafter, as throughout the literature of general relativity, a real time coordinate is used, x 0 = t = c t conv x 0 = t = c t conv  x^(0)=t=ct_("conv ")x^{0}=t=c t_{\text {conv }}x0=t=ctconv  (superscript 0 rather than 4 to avoid any possibility of confusion with the imaginary time coordinate).
The components w α w α w^(alpha)w^{\alpha}wα of any other vector w w w\boldsymbol{w}w in this frame are similarly defined as the coefficients in such an expansion,
(2.4) w = w α e α (2.4) w = w α e α {:(2.4)w=w^(alpha)e_(alpha):}\begin{equation*} \boldsymbol{w}=w^{\alpha} \boldsymbol{e}_{\alpha} \tag{2.4} \end{equation*}(2.4)w=wαeα
Notice: the subscript α α alpha\alphaα on e α e α e_(alpha)\boldsymbol{e}_{\alpha}eα tells which vector, not which component!

§2.4. THE METRIC TENSOR

The metric tensor, one recalls from part IV of Box 1.3, is a machine for calculating the squared length of a single vector, or the scalar product of two different vectors.
Expansion of vector in terms of basis
Figure 2.3.
The 4 -velocity of a particle in flat spacetime. The 4 -velocity u u u\boldsymbol{u}u is the unit vector (arrow) tangent to the particle's world line-one tangent vector for each event on the world line. In a specific Lorentz coordinate system, there are basis vectors of unit length, which point along the four coordinate axes: e 0 , e 1 , e 2 , e 3 e 0 , e 1 , e 2 , e 3 e_(0),e_(1),e_(2),e_(3)\boldsymbol{e}_{0}, \boldsymbol{e}_{1}, \boldsymbol{e}_{2}, \boldsymbol{e}_{3}e0,e1,e2,e3. The 4-velocity, like any vector, can be expressed as a sum of components along the basis vectors:
u = u 0 e 0 + u 1 e 1 + u 2 e 2 + u 3 e 3 = u α e α . u = u 0 e 0 + u 1 e 1 + u 2 e 2 + u 3 e 3 = u α e α . u=u^(0)e_(0)+u^(1)e_(1)+u^(2)e_(2)+u^(3)e_(3)=u^(alpha)e_(alpha).\boldsymbol{u}=u^{0} \mathbf{e}_{0}+u^{1} \boldsymbol{e}_{1}+u^{2} \boldsymbol{e}_{2}+u^{3} \boldsymbol{e}_{3}=u^{\alpha} \boldsymbol{e}_{\alpha} .u=u0e0+u1e1+u2e2+u3e3=uαeα.
Metric defined as machine for computing scalar products of vectors
More precisely, the metric tensor g g g\boldsymbol{g}g is a machine with two slots for inserting vectors
Upon insertion, the machine spews out a real number:
(2.6) g ( u , v ) = "scalar product of u and v ," also denoted u v . g ( u , u ) = "squared length of u , " also denoted u 2 . (2.6) g ( u , v ) =  "scalar product of  u  and  v ," also denoted  u v . g ( u , u ) =  "squared length of  u , "  also denoted  u 2 . {:[(2.6)g(u","v)=" "scalar product of "u" and "v"," also denoted "u*v.],[g(u","u)=" "squared length of "u",""" also denoted "u^(2).]:}\begin{align*} & \boldsymbol{g}(\boldsymbol{u}, \boldsymbol{v})=\text { "scalar product of } \boldsymbol{u} \text { and } \boldsymbol{v} \text {," also denoted } \boldsymbol{u} \cdot \boldsymbol{v} . \tag{2.6}\\ & \boldsymbol{g}(\boldsymbol{u}, \boldsymbol{u})=\text { "squared length of } \boldsymbol{u}, " \text { also denoted } \boldsymbol{u}^{2} . \end{align*}(2.6)g(u,v)= "scalar product of u and v," also denoted uv.g(u,u)= "squared length of u," also denoted u2.
Moreover, this number is independent of the order in which the vectors are inserted ("symmetry of metric tensor"),
(2.7) g ( u , v ) = g ( v , u ) (2.7) g ( u , v ) = g ( v , u ) {:(2.7)g(u","v)=g(v","u):}\begin{equation*} \boldsymbol{g}(\boldsymbol{u}, \boldsymbol{v})=\boldsymbol{g}(\boldsymbol{v}, \boldsymbol{u}) \tag{2.7} \end{equation*}(2.7)g(u,v)=g(v,u)
and it is linear in the vectors inserted
(2.8) g ( a u + b v , w ) = g ( w , a u + b v ) = a g ( u , w ) + b g ( v , w ) (2.8) g ( a u + b v , w ) = g ( w , a u + b v ) = a g ( u , w ) + b g ( v , w ) {:(2.8)g(au+bv","w)=g(w","au+bv)=ag(u","w)+bg(v","w):}\begin{equation*} \boldsymbol{g}(a \boldsymbol{u}+b \mathbf{v}, \boldsymbol{w})=\boldsymbol{g}(\boldsymbol{w}, a \boldsymbol{u}+b \boldsymbol{v})=a \boldsymbol{g}(\boldsymbol{u}, \boldsymbol{w})+b \boldsymbol{g}(\boldsymbol{v}, \boldsymbol{w}) \tag{2.8} \end{equation*}(2.8)g(au+bv,w)=g(w,au+bv)=ag(u,w)+bg(v,w)
Because the metric "machine" is linear, one can calculate its output, for any input,
as follows, if one knows only what it does to the basis vectors e α e α e_(alpha)\boldsymbol{e}_{\alpha}eα of a Lorentz frame.
(1) Define the symbols ("metric coefficients") η α β η α β eta_(alpha beta)\eta_{\alpha \beta}ηαβ by
(2.9) η α β g ( e α , e β ) = e α e β (2.9) η α β g e α , e β = e α e β {:(2.9)eta_(alpha beta)-=g(e_(alpha),e_(beta))=e_(alpha)*e_(beta):}\begin{equation*} \eta_{\alpha \beta} \equiv \boldsymbol{g}\left(\boldsymbol{e}_{\alpha}, \boldsymbol{e}_{\beta}\right)=\boldsymbol{e}_{\alpha} \cdot \boldsymbol{e}_{\beta} \tag{2.9} \end{equation*}(2.9)ηαβg(eα,eβ)=eαeβ
(2) Calculate their numerical values from the known squared length of the separation vector ξ = Δ x α e α ξ = Δ x α e α xi=Deltax^(alpha)e_(alpha)\boldsymbol{\xi}=\Delta x^{\alpha} \boldsymbol{e}_{\alpha}ξ=Δxαeα between two events:
( Δ s ) 2 = ( Δ x 0 ) 2 + ( Δ x 1 ) 2 + ( Δ x 2 ) 2 + ( Δ x 3 ) 2 = g ( Δ x α e α , Δ x β e β ) = Δ x α Δ x β g ( e α , e β ) = Δ x α Δ x β η α β for every choice of Δ x α (2.10) η α β 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 in any Lorentz frame. ( Δ s ) 2 = Δ x 0 2 + Δ x 1 2 + Δ x 2 2 + Δ x 3 2 = g Δ x α e α , Δ x β e β = Δ x α Δ x β g e α , e β = Δ x α Δ x β η α β  for every choice of  Δ x α (2.10) η α β 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1  in any Lorentz frame.  {:[(Delta s)^(2)=-(Deltax^(0))^(2)+(Deltax^(1))^(2)+(Deltax^(2))^(2)+(Deltax^(3))^(2)],[=g(Deltax^(alpha)e_(alpha),Deltax^(beta)e_(beta))=Deltax^(alpha)Deltax^(beta)g(e_(alpha),e_(beta))],[=Deltax^(alpha)Deltax^(beta)eta_(alpha beta)quad" for every choice of "Deltax^(alpha)],[(2.10) Longrightarrow||eta_(alpha beta)||-=||[-1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]||" in any Lorentz frame. "]:}\begin{align*} (\Delta s)^{2} & =-\left(\Delta x^{0}\right)^{2}+\left(\Delta x^{1}\right)^{2}+\left(\Delta x^{2}\right)^{2}+\left(\Delta x^{3}\right)^{2} \\ & =\boldsymbol{g}\left(\Delta x^{\alpha} \boldsymbol{e}_{\alpha}, \Delta x^{\beta} \boldsymbol{e}_{\beta}\right)=\Delta x^{\alpha} \Delta x^{\beta} \boldsymbol{g}\left(\boldsymbol{e}_{\alpha}, \boldsymbol{e}_{\beta}\right) \\ & =\Delta x^{\alpha} \Delta x^{\beta} \eta_{\alpha \beta} \quad \text { for every choice of } \Delta x^{\alpha} \\ & \Longrightarrow\left\|\eta_{\alpha \beta}\right\| \equiv\left\|\begin{array}{rrrrr} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right\| \text { in any Lorentz frame. } \tag{2.10} \end{align*}(Δs)2=(Δx0)2+(Δx1)2+(Δx2)2+(Δx3)2=g(Δxαeα,Δxβeβ)=ΔxαΔxβg(eα,eβ)=ΔxαΔxβηαβ for every choice of Δxα(2.10)ηαβ1000010000100001 in any Lorentz frame. 
(3) Calculate the scalar product of any two vectors u u u\boldsymbol{u}u and v v v\boldsymbol{v}v from
u v = g ( u , v ) = g ( u α e α , v β e β ) = u α v β g ( e α , e β ) ; (2.11) u v = u α v β η α β = u 0 v 0 + u 1 v 1 + u 2 v 2 + u 3 v 3 . u v = g ( u , v ) = g u α e α , v β e β = u α v β g e α , e β ; (2.11) u v = u α v β η α β = u 0 v 0 + u 1 v 1 + u 2 v 2 + u 3 v 3 . {:[u*v=g(u","v)=g(u^(alpha)e_(alpha),v^(beta)e_(beta))=u^(alpha)v^(beta)g(e_(alpha),e_(beta));],[(2.11)u*v=u^(alpha)v^(beta)eta_(alpha beta)=-u^(0)v^(0)+u^(1)v^(1)+u^(2)v^(2)+u^(3)v^(3).]:}\begin{align*} & \boldsymbol{u} \cdot \boldsymbol{v}=\boldsymbol{g}(\boldsymbol{u}, \boldsymbol{v})=\boldsymbol{g}\left(u^{\alpha} \boldsymbol{e}_{\alpha}, v^{\beta} \boldsymbol{e}_{\beta}\right)=u^{\alpha} v^{\beta} \boldsymbol{g}\left(\boldsymbol{e}_{\alpha}, \boldsymbol{e}_{\beta}\right) ; \\ & \boldsymbol{u} \cdot \boldsymbol{v}=u^{\alpha} v^{\beta} \eta_{\alpha \beta}=-u^{0} v^{0}+u^{1} v^{1}+u^{2} v^{2}+u^{3} v^{3} . \tag{2.11} \end{align*}uv=g(u,v)=g(uαeα,vβeβ)=uαvβg(eα,eβ);(2.11)uv=uαvβηαβ=u0v0+u1v1+u2v2+u3v3.
That one can classify directions and vectors in spacetime into "timelike" (negative squared length), "spacelike"(positive squared length), and "null" or "lightlike" (zero squared length) is made possible by the negative sign on the metric coefficient η 00 η 00 eta_(00)\eta_{00}η00.
Box 2.2 shows applications of the above ideas and notation to two elementary problems in special relativity theory.

§2.5. DIFFERENTIAL FORMS

Vectors and the metric tensor are geometric objects that are already familiar from Chapter 1 and from elementary courses in special relativity. Not so familiar, yet equally important, is a third geometric object: the "differential form" or " 1 -form."
Consider the 4-momentum p p p\boldsymbol{p}p of a particle, an electron, for example. To spell out one concept of momentum, start with the 4 -velocity, u = d P / d τ u = d P / d τ u=dP//d tau\boldsymbol{u}=d \mathscr{P} / d \tauu=dP/dτ, of this electron ("spacetime displacement per unit of proper time along a straightline approximation of the world line"). This is a vector of unit length. Multiply by the mass m m mmm of the particle to obtain the momentum vector
p = m u . p = m u . p=mu.\boldsymbol{p}=m \boldsymbol{u} .p=mu.
But physics gives also quite another idea of momentum. It associates a de Broglie wave with each particle. Moreover, this wave has the most direct possible physical significance. Diffract this wave from a crystal lattice. From the pattern of diffraction, one can determine not merely the length of the de Broglie waves, but also the pattern in space made by surfaces of equal, integral phase ϕ = 7 , ϕ = 8 , ϕ = 9 , ϕ = 7 , ϕ = 8 , ϕ = 9 , phi=7,phi=8,phi=9,dots\phi=7, \phi=8, \phi=9, \ldotsϕ=7,ϕ=8,ϕ=9,. This
Scalar products computed from components of vectors
The 1 -form illustrated by de Broglie waves

Box 2.2 WORKED EXERCISES USING THE METRIC

Exercise: Show that the squared length of a test particle's 4 -velocity u u u\boldsymbol{u}u is -1 . Solution: In any Lorentz frame, using the components (2.2), one calculates as follows
u 2 = g ( u , u ) = u α u β η α β = ( u 0 ) 2 + ( u 1 ) 2 + ( u 2 ) 2 + ( u 3 ) 2 = 1 1 v 2 + v 2 1 v 2 = 1 . u 2 = g ( u , u ) = u α u β η α β = u 0 2 + u 1 2 + u 2 2 + u 3 2 = 1 1 v 2 + v 2 1 v 2 = 1 . {:[u^(2)=g(u","u)=u^(alpha)u^(beta)eta_(alpha beta)=-(u^(0))^(2)+(u^(1))^(2)+(u^(2))^(2)+(u^(3))^(2)],[=-(1)/(1-v^(2))+(v^(2))/(1-v^(2))=-1.]:}\begin{aligned} \boldsymbol{u}^{2} & =\boldsymbol{g}(\boldsymbol{u}, \boldsymbol{u})=u^{\alpha} u^{\beta} \eta_{\alpha \beta}=-\left(u^{0}\right)^{2}+\left(u^{1}\right)^{2}+\left(u^{2}\right)^{2}+\left(u^{3}\right)^{2} \\ & =-\frac{1}{1-\boldsymbol{v}^{2}}+\frac{\boldsymbol{v}^{2}}{1-\boldsymbol{v}^{2}}=-1 . \end{aligned}u2=g(u,u)=uαuβηαβ=(u0)2+(u1)2+(u2)2+(u3)2=11v2+v21v2=1.
Exercise: Show that the rest mass of a particle is related to its energy and momentum by the famous equation
( m c 2 ) 2 = E 2 ( p c ) 2 m c 2 2 = E 2 ( p c ) 2 (mc^(2))^(2)=E^(2)-(pc)^(2)\left(m c^{2}\right)^{2}=E^{2}-(\boldsymbol{p} c)^{2}(mc2)2=E2(pc)2
or, equivalently (geometrized units!),
m 2 = E 2 p 2 m 2 = E 2 p 2 m^(2)=E^(2)-p^(2)m^{2}=E^{2}-p^{2}m2=E2p2
First Solution: The 4-momentum is defined by p = m u p = m u p=mu\boldsymbol{p}=m \boldsymbol{u}p=mu, where u u u\boldsymbol{u}u is the 4 -velocity and m m mmm is the rest mass. Consequently, its squared length is
p 2 = m 2 u 2 = m 2 = ( m u 0 ) 2 + m 2 u 2 = m 2 1 v 2 + m 2 v 2 1 v 2 E 2 p 2 p 2 = m 2 u 2 = m 2 = m u 0 2 + m 2 u 2 = m 2 1 v 2 + m 2 v 2 1 v 2 E 2 p 2 {:[p^(2)=m^(2)u^(2)=-m^(2)],[=-(mu^(0))^(2)+m^(2)u^(2)=-(m^(2))/(1-v^(2))+(m^(2)v^(2))/(1-v^(2))],[E^(2)_(p^(2))]:}\begin{aligned} \boldsymbol{p}^{2} & =m^{2} \boldsymbol{u}^{2}=-m^{2} \\ & =-\left(m u^{0}\right)^{2}+m^{2} \boldsymbol{u}^{2}=-\frac{m^{2}}{1-\boldsymbol{v}^{2}}+\frac{m^{2} \boldsymbol{v}^{2}}{1-\boldsymbol{v}^{2}} \\ & \mathrm{E}^{2} \underset{\boldsymbol{p}^{2}}{ } \end{aligned}p2=m2u2=m2=(mu0)2+m2u2=m21v2+m2v21v2E2p2
Second Solution: In the frame of the observer, where E E EEE and p p ppp are measured, the 4 -momentum splits into time and space parts as
p 0 = E , p 1 e 1 + p 2 e 2 + p 3 e 3 = p p 0 = E , p 1 e 1 + p 2 e 2 + p 3 e 3 = p p^(0)=E,quadp^(1)e_(1)+p^(2)e_(2)+p^(3)e_(3)=pp^{0}=E, \quad p^{1} \boldsymbol{e}_{1}+p^{2} \boldsymbol{e}_{2}+p^{3} \boldsymbol{e}_{3}=\boldsymbol{p}p0=E,p1e1+p2e2+p3e3=p
hence, its squared length is
p 2 = E 2 + p 2 . p 2 = E 2 + p 2 . p^(2)=-E^(2)+p^(2).\boldsymbol{p}^{2}=-E^{2}+\boldsymbol{p}^{2} .p2=E2+p2.
But in the particle's rest frame, p p p\boldsymbol{p}p splits as
p 0 = m , p 1 = p 2 = p 3 = 0 ; p 0 = m , p 1 = p 2 = p 3 = 0 ; p^(0)=m,quadp^(1)=p^(2)=p^(3)=0;p^{0}=m, \quad p^{1}=p^{2}=p^{3}=0 ;p0=m,p1=p2=p3=0;
hence, its squared length is p 2 = m 2 p 2 = m 2 p^(2)=-m^(2)\boldsymbol{p}^{2}=-m^{2}p2=m2. But the squared length is a geometric object defined independently of any coordinate system; so it must be the same by whatever means one calculates it:
p 2 = m 2 = E 2 p 2 . p 2 = m 2 = E 2 p 2 . -p^(2)=m^(2)=E^(2)-p^(2).-\boldsymbol{p}^{2}=m^{2}=E^{2}-\boldsymbol{p}^{2} .p2=m2=E2p2.
Figure 2.4.
The vector separation v = P P 0 v = P P 0 v=P-P_(0)\boldsymbol{v}=\mathscr{P}-\mathscr{P}_{0}v=PP0 between two neighboring events P 0 P 0 P_(0)\mathscr{P}_{0}P0 and P P P\mathscr{P}P; a 1 -form σ σ sigma\boldsymbol{\sigma}σ; and the piercing of σ σ sigma\boldsymbol{\sigma}σ by v v v\boldsymbol{v}v to give the number
σ , v = ( number of surfaces pierced ) = 4.4 σ , v = (  number of surfaces pierced  ) = 4.4 (:sigma,v:)=(" number of surfaces pierced ")=4.4\langle\sigma, v\rangle=(\text { number of surfaces pierced })=4.4σ,v=( number of surfaces pierced )=4.4
( 4.4 "bongs of bell"). When σ σ sigma\boldsymbol{\sigma}σ is made of surfaces of constant phase, ϕ = 17 , ϕ = 18 , ϕ = 19 , ϕ = 17 , ϕ = 18 , ϕ = 19 , phi=17,phi=18,phi=19,dots\phi=17, \phi=18, \phi=19, \ldotsϕ=17,ϕ=18,ϕ=19, of the de Broglie wave for an electron, then σ , v σ , v (:sigma,v:)\langle\sigma, \boldsymbol{v}\rangleσ,v is the phase difference between the events P 0 P 0 P_(0)\mathscr{P}_{0}P0 and P P P\mathscr{P}P. Note that σ σ sigma\sigmaσ is not fully specified by its surfaces; an orientation is also necessary. Which direction from surface to surface is "positive"; i.e., in which direction does ϕ ϕ phi\phiϕ increase?
pattern of surfaces, given a name " K ~ K ~ widetilde(K)\widetilde{\boldsymbol{K}}K~," provides the simplest illustration one can easily find for a 1 -form.
The pattern of surfaces in spacetime made by such a 1 -form: what is it good for? Take two nearby points in spacetime, P P P\mathscr{P}P and P 0 P 0 P_(0)\mathscr{P}_{0}P0. Run an arrow v = P P 0 v = P P 0 v=P-P_(0)\boldsymbol{v}=\mathscr{P}-\mathscr{P}_{0}v=PP0 from
P 0 P 0 P_(0)\mathscr{P}_{0}P0 to P P P\mathscr{P}P. It will pierce a certain number of the de Broglie wave's surfaces of integral phase, with a bong of an imaginary bell at each piercing. The number of surfaces pierced (number of "bongs of bell") is denoted
in this example it equals the phase difference between tail ( P 0 ) P 0 (P_(0))\left(\mathscr{P}_{0}\right)(P0) and tip ( P ) ( P ) (P)(\mathscr{P})(P) of v v v\boldsymbol{v}v,
k ~ , v = ϕ ( P ) ϕ ( P 0 ) . k ~ , v = ϕ ( P ) ϕ P 0 . (: widetilde(k),v:)=phi(P)-phi(P_(0)).\langle\widetilde{\boldsymbol{k}}, \boldsymbol{v}\rangle=\phi(\mathscr{P})-\phi\left(\mathscr{P}_{0}\right) .k~,v=ϕ(P)ϕ(P0).

See Figure 2.4.

Normally neither P 0 P 0 P_(0)\mathscr{P}_{0}P0 nor P P P\mathscr{P}P will lie at a point of integral phase. Therefore one
can and will imagine, as uniformly interpolated between the surfaces of integral
phase, an infinitude of surfaces with all the intermediate phase values. With their
can and will imagine, as uniformly interpolated between the surfaces of integral
phase, an infinitude of surfaces with all the intermediate phase values. With their aid, the precise value of k ~ , v = ϕ ( P ) ϕ ( P 0 ) k ~ , v = ϕ ( P ) ϕ P 0 (: widetilde(k),v:)=phi(P)-phi(P_(0))\langle\widetilde{\boldsymbol{k}}, \boldsymbol{v}\rangle=\phi(\mathscr{P})-\phi\left(\mathscr{P}_{0}\right)k~,v=ϕ(P)ϕ(P0) can be determined.
To make the mathematics simple, regard k ~ k ~ widetilde(k)\widetilde{\boldsymbol{k}}k~ not as the global pattern of de Brogliewave surfaces, but as a local pattern near a specific point in spacetime. Just as the vector u = d P / d τ u = d P / d τ u=dP//d tau\boldsymbol{u}=d \mathscr{P} / d \tauu=dP/dτ represents the local behavior of a particle's world line (linear approximation to curved line in general), so the 1-form k ~ k ~ widetilde(k)\widetilde{\boldsymbol{k}}k~ represents the local form
Vector pierces 1-form
pieced (n mer of is or
Norn
To make the mathematics simple, regard k ~ k ~ tilde(k)\tilde{k}k~ not as the global paltern of de Brogic-
The 1 -form viewed as family of flat, equally spaced surfaces
Figure 2.5.
This is a dual-purpose figure. (a) It illustrates the de Broglie wave 1 -form k ~ k ~ widetilde(k)\widetilde{\boldsymbol{k}}k~ at an event P 0 P 0 P_(0)\mathscr{P}_{0}P0 (family of equally spaced, flat surfaces, or "hyperplanes" approximating the surfaces of constant phase). (b) It illustrates the gradient d ϕ d ϕ d phi\boldsymbol{d} \phidϕ of the function ϕ ϕ phi\phiϕ (concept defined in $ 2.6 $ 2.6 $2.6\$ 2.6$2.6 ), which is the same oriented family of flat surfaces
k ~ = d ϕ . k ~ = d ϕ . widetilde(k)=d phi.\widetilde{\boldsymbol{k}}=\boldsymbol{d} \phi .k~=dϕ.
At different events, k ~ = d ϕ k ~ = d ϕ widetilde(k)=d phi\widetilde{\boldsymbol{k}}=\boldsymbol{d} \phik~=dϕ is different-different orientation of surfaces and different spacing. The change in ϕ ϕ phi\phiϕ between the tail and tip of the very short vector v v v\boldsymbol{v}v is equal to the number of surfaces of d ϕ d ϕ d phi\boldsymbol{d} \phidϕ pierced by v , d ϕ , v v , d ϕ , v v,(:d phi,v:)\boldsymbol{v},\langle\boldsymbol{d} \phi, \boldsymbol{v}\ranglev,dϕ,v; it equals -0.5 in this figure.
of the de Broglie wave's surfaces (linear approximation; surfaces flat and equally spaced; see Figure 2.5).
Regard the 1 -form k ~ k ~ widetilde(k)\widetilde{\boldsymbol{k}}k~ as a machine into which vectors are inserted, and from which numbers emerge. Insertion of v v v\boldsymbol{v}v produces as output k ~ , v k ~ , v (: widetilde(k),v:)\langle\widetilde{\boldsymbol{k}}, \boldsymbol{v}\ranglek~,v. Since the surfaces of k ~ k ~ widetilde(k)\widetilde{\boldsymbol{k}}k~ are flat and equally spaced, the output is a linear function of the input:
(2.12a) k ~ , a u + b v = a k ~ , u + b k ~ , v . (2.12a) k ~ , a u + b v = a k ~ , u + b k ~ , v . {:(2.12a)(: widetilde(k)","au+bv:)=a(: widetilde(k)","u:)+b(: widetilde(k)","v:).:}\begin{equation*} \langle\widetilde{\boldsymbol{k}}, a \boldsymbol{u}+b \boldsymbol{v}\rangle=a\langle\widetilde{\boldsymbol{k}}, \boldsymbol{u}\rangle+b\langle\widetilde{\boldsymbol{k}}, \boldsymbol{v}\rangle . \tag{2.12a} \end{equation*}(2.12a)k~,au+bv=ak~,u+bk~,v.
The 1 -form viewed as linear function of vectors
Figure 2.6.
The addition of two 1 -forms, α α alpha\boldsymbol{\alpha}α and β β beta\boldsymbol{\beta}β, to produce the 1 -form σ σ sigma\boldsymbol{\sigma}σ. Required is a pictorial construction that starts from the surfaces of α α alpha\boldsymbol{\alpha}α and β β beta\boldsymbol{\beta}β, e.g., α , P P 0 = 1 , 0 , 1 , 2 , α , P P 0 = 1 , 0 , 1 , 2 , (:alpha,P-P_(0):)=cdots-1,0,1,2,dots\left\langle\boldsymbol{\alpha}, \mathscr{P}-\mathscr{P}_{0}\right\rangle=\cdots-1,0,1,2, \ldotsα,PP0=1,0,1,2,, and constructs those of σ = α + β σ = α + β sigma=alpha+beta\sigma=\boldsymbol{\alpha}+\boldsymbol{\beta}σ=α+β. Such a construction, based on linearity (2.12b) of the addition process, is as follows. (1) Pick several vectors u , v , u , v , u,v,dots\boldsymbol{u}, \boldsymbol{v}, \ldotsu,v, that lie parallel to the surfaces of β β beta\boldsymbol{\beta}β (no piercing!), but pierce precisely 3 surfaces of α α alpha\boldsymbol{\alpha}α; each of these must then pierce precisely 3 surfaces of σ σ sigma\boldsymbol{\sigma}σ :
σ , u = α + β , u = a , u = 3 . σ , u = α + β , u = a , u = 3 . (:sigma,u:)=(:alpha+beta,u:)=(:a,u:)=3.\langle\boldsymbol{\sigma}, \boldsymbol{u}\rangle=\langle\boldsymbol{\alpha}+\boldsymbol{\beta}, \boldsymbol{u}\rangle=\langle\boldsymbol{a}, \boldsymbol{u}\rangle=3 .σ,u=α+β,u=a,u=3.
(2) Pick several other vectors w , w , w,dots\boldsymbol{w}, \ldotsw, that lie parallel to the surfaces of α α alpha\boldsymbol{\alpha}α but pierce precisely 3 surfaces of β β beta\boldsymbol{\beta}β; these will also pierce precisely 3 surfaces of σ σ sigma\boldsymbol{\sigma}σ. (3) Construct that unique family of equally spaced surfaces in which u , v , , w , u , v , , w , u,v,dots,w,dots\boldsymbol{u}, \mathbf{v}, \ldots, \boldsymbol{w}, \ldotsu,v,,w, all have their tails on one surface and their tips on the third succeeding surface.
Sometimes 1-forms are denoted by boldface, sans-serif Latin letters with tildes over them, e.g., k ~ k ~ widetilde(k)\widetilde{\boldsymbol{k}}k~; but more often by boldface Greek letters, e.g., α , β , σ α , β , σ alpha,beta,sigma\boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{\sigma}α,β,σ. The output of a 1 -form σ σ sigma\boldsymbol{\sigma}σ, when a vector u u u\boldsymbol{u}u is inserted, is called "the value of σ σ sigma\boldsymbol{\sigma}σ on u u u\boldsymbol{u}u " or "the contraction of σ σ sigma\boldsymbol{\sigma}σ with u u u\boldsymbol{u}u."
Also, 1-forms, like any other kind of function, can be added. The 1-form a α + b β a α + b β a alpha+b betaquada \boldsymbol{\alpha}+b \boldsymbol{\beta} \quadaα+bβ Addition of 1 -forms is that machine (family of surfaces) which puts out the following number when a vector u u u\boldsymbol{u}u is put in:
(2.12b) a α + b β , u = a α , u + b β , u . (2.12b) a α + b β , u = a α , u + b β , u . {:(2.12b)(:a alpha+b beta","u:)=a(:alpha","u:)+b(:beta","u:).:}\begin{equation*} \langle a \boldsymbol{\alpha}+b \boldsymbol{\beta}, \boldsymbol{u}\rangle=a\langle\boldsymbol{\alpha}, \boldsymbol{u}\rangle+b\langle\boldsymbol{\beta}, \boldsymbol{u}\rangle . \tag{2.12b} \end{equation*}(2.12b)aα+bβ,u=aα,u+bβ,u.
Figure 2.6 depicts this addition in terms of surfaces.
One can verify that the set of all 1 -forms at a given event is a "vector space" in the abstract, algebraic sense of the term.
Return to a particle and its de Broglie wave. Just as the arrow p = m d P / d τ p = m d P / d τ p=mdP//d tau\boldsymbol{p}=m d \mathscr{P} / d \taup=mdP/dτ represents the best linear approximation to the particle's actual world line near P 0 P 0 P_(0)\mathscr{P}_{0}P0, so the flat surfaces of the 1 -form k ~ k ~ widetilde(k)\widetilde{\boldsymbol{k}}k~ provide the best linear approximation to the curved surfaces of the particle's de Broglie wave, and k ~ k ~ widetilde(k)\widetilde{\boldsymbol{k}}k~ itself is the linear function that best approximates the de Broglie phase ϕ ϕ phi\phiϕ near P 0 P 0 P_(0)\mathscr{P}_{0}P0 :
(2.13) ϕ ( P ) = ϕ ( P 0 ) + k ~ , P P 0 + terms of higher order in ( P P 0 ) . (2.13) ϕ ( P ) = ϕ P 0 + k ~ , P P 0 +  terms of higher order in  P P 0 . {:[(2.13)phi(P)=phi(P_(0))+(:( widetilde(k)),P-P_(0):)],[+" terms of higher order in "(P-P_(0)).]:}\begin{align*} \phi(\mathscr{P})=\phi\left(\mathscr{P}_{0}\right) & +\left\langle\widetilde{\boldsymbol{k}}, \mathscr{P}-\mathscr{P}_{0}\right\rangle \tag{2.13}\\ & + \text { terms of higher order in }\left(\mathscr{P}-\mathscr{P}_{0}\right) . \end{align*}(2.13)ϕ(P)=ϕ(P0)+k~,PP0+ terms of higher order in (PP0).
Figure 2.7.
Several vectors, A , B , C , D , E A , B , C , D , E A,B,C,D,E\boldsymbol{A}, \boldsymbol{B}, \boldsymbol{C}, \boldsymbol{D}, \boldsymbol{E}A,B,C,D,E, and corresponding 1 -forms A ~ , B ~ , C ~ , D ~ , E ~ A ~ , B ~ , C ~ , D ~ , E ~ widetilde(A), widetilde(B), widetilde(C), widetilde(D), widetilde(E)\widetilde{\boldsymbol{A}}, \widetilde{\boldsymbol{B}}, \widetilde{\boldsymbol{C}}, \widetilde{\boldsymbol{D}}, \widetilde{\boldsymbol{E}}A~,B~,C~,D~,E~. The process of drawing U ~ U ~ widetilde(U)\widetilde{\boldsymbol{U}}U~ corresponding to a given vector U U U\boldsymbol{U}U is quite simple. (1) Orient the surfaces of U ~ U ~ widetilde(U)\widetilde{\boldsymbol{U}}U~ orthogonal to the vector U U U\boldsymbol{U}U. (Why? Because any vector V V V\boldsymbol{V}V that is perpendicular to U U U\boldsymbol{U}U must pierce no surfaces of U ~ U ~ widetilde(U)\widetilde{\boldsymbol{U}}U~ ( 0 = U V = U ~ , V 0 = U V = U ~ , V 0=U*V=(: widetilde(U),V:)0=\boldsymbol{U} \cdot \boldsymbol{V}=\langle\widetilde{\boldsymbol{U}}, \boldsymbol{V}\rangle0=UV=U~,V ) and must therefore lie in a surface of U ~ U ~ widetilde(U)\widetilde{\boldsymbol{U}}U~.) (2) Space the surfaces of U ~ U ~ widetilde(U)\widetilde{\boldsymbol{U}}U~ so that the number of surfaces pierced by some arbitrary vector V V V\boldsymbol{V}V (e.g., V = U V = U V=U\boldsymbol{V}=\boldsymbol{U}V=U ) is equal to V U V U V*U\boldsymbol{V} \cdot \boldsymbol{U}VU.
Note that in the figure the surfaces of B ~ B ~ widetilde(B)\widetilde{\boldsymbol{B}}B~ are, indeed, orthogonal to B B B\boldsymbol{B}B; those of C ~ C ~ widetilde(C)\widetilde{\boldsymbol{C}}C~ are, indeed, orthogonal to C C C\boldsymbol{C}C, etc. If they do not look so, that is because the reader is attributing Euclidean geometry, not Lorentz geometry, to the spacetime diagram. He should recall, for example, that because C C C\boldsymbol{C}C is a null vector, it is orthogonal to itself ( C C = 0 ) ( C C = 0 ) (C*C=0)(\boldsymbol{C} \cdot \boldsymbol{C}=0)(CC=0), so it must itself lie in a surface of the 1 -form C C C\boldsymbol{C}C. Confused readers may review spacetime diagrams in a more elementary text, e.g., Taylor and Wheeler (1966).
Physical correspondence between 1 -forms and vectors
Mathematical correspondence between 1 -forms and vectors
Actually, the de Broglie 1 -form k ~ k ~ widetilde(k)\widetilde{\boldsymbol{k}}k~ and the momentum vector p p p\boldsymbol{p}p contain precisely the same information, both physically (via quantum theory) and mathematically. To see their relationship. relabel the surfaces of κ ~ κ ~ widetilde(kappa)\widetilde{\boldsymbol{\kappa}}κ~ by × × ℏxx\hbar \times× phase, thereby obtaining the "momentum 1 -form" p ~ p ~ widetilde(p)\widetilde{\boldsymbol{p}}p~. Pierce this 1 -form with any vector v v v\boldsymbol{v}v, and find the result (exercise 2.1) that
(2.14) p v = p ~ , v . (2.14) p v = p ~ , v . {:(2.14)p*v=(: widetilde(p)","v:).:}\begin{equation*} \boldsymbol{p} \cdot \boldsymbol{v}=\langle\widetilde{\boldsymbol{p}}, \boldsymbol{v}\rangle . \tag{2.14} \end{equation*}(2.14)pv=p~,v.
In words: the projection of v v v\boldsymbol{v}v on the 4 -momentum vector p p p\boldsymbol{p}p equals the number of surfaces it pierces in the 4 -momentum 1 -form p ~ p ~ widetilde(p)\widetilde{\boldsymbol{p}}p~. Examples: Vectors v v v\boldsymbol{v}v lying in a surface of p ~ p ~ widetilde(p)\widetilde{\boldsymbol{p}}p~ (no piercing) are perpendicular to p p p\boldsymbol{p}p (no projection); p p p\boldsymbol{p}p itself pierces p 2 = m 2 p 2 = m 2 p^(2)=-m^(2)\boldsymbol{p}^{2}=-m^{2}p2=m2 surfaces of p ~ p ~ widetilde(p)\widetilde{\boldsymbol{p}}p~.
Corresponding to any vector p p p\boldsymbol{p}p there exists a unique 1 -form (linear function of vectors) p ~ p ~ widetilde(p)\widetilde{\boldsymbol{p}}p~ defined by equation (2.14). And corresponding to any 1-form p ~ p ~ widetilde(p)\widetilde{\boldsymbol{p}}p~, there exists a unique vector p p p\boldsymbol{p}p defined by its projections on all other vectors, by equation (2.14). Figure 2.7 shows several vectors and their corresponding 1 -forms.
A single physical quantity can be described equally well by a vector p p p\boldsymbol{p}p or by the corresponding 1 -form p ~ p ~ widetilde(p)\widetilde{\boldsymbol{p}}p~. Sometimes the vector description is the simplest and most natural; sometimes the 1-form description is nicer. Example: Consider a 1 -form representing the march of Lorentz coordinate time toward the future-surfaces x 0 = , 7 , 8 , 9 , x 0 = , 7 , 8 , 9 , x^(0)=dots,7,8,9,dotsx^{0}=\ldots, 7,8,9, \ldotsx0=,7,8,9, The corresponding vector points toward the past [see Figure 2.7 or equation (2.14)]; its description of the forward march of time is not so nice!
One often omits the tilde from the 1 -form p ~ p ~ widetilde(p)\widetilde{\boldsymbol{p}}p~ corresponding to a vector p p p\boldsymbol{p}p, and uses the same symbol p p p\boldsymbol{p}p for both. Such practice is justified by the unique correspondence (both mathematical and physical) between p ~ p ~ widetilde(p)\widetilde{\boldsymbol{p}}p~ and p p p\boldsymbol{p}p.

Exercise 2.1.

EXERCISE

Show that equation (2.14) is in accord with the quantum-mechanical properties of a de Broglie wave,
ψ = e i ϕ = exp [ i ( k x ω t ) ] . ψ = e i ϕ = exp [ i ( k x ω t ) ] . psi=e^(i phi)=exp[i(k*x-omega t)].\psi=e^{i \phi}=\exp [i(\boldsymbol{k} \cdot x-\omega t)] .ψ=eiϕ=exp[i(kxωt)].

§2.6. GRADIENTS AND DIRECTIONAL DERIVATIVES

There is no simpler 1-form than the gradient, "df," of a function f f fff. Gradient a 1-form? How so? Hasn't one always known the gradient as a vector? Yes, indeed, but only because one was not familiar with the more appropriate 1 -form concept. The more familiar gradient is the vector corresponding, via equation (2.14), to the 1 -form gradient. The hyperplanes representing d f d f df\boldsymbol{d} fdf at a point P 0 P 0 P_(0)\mathscr{P}_{0}P0 are just the level surfaces of f f fff itself, except for flattening and adjustment to equal spacing (Figure 2.5; identify f f fff here with ϕ ϕ phi\phiϕ there). More precisely, they are the level surfaces of the linear function that approximates f f fff in an infinitesimal neighborhood of P 0 P 0 P_(0)\mathscr{\mathscr { P }}_{0}P0.
Why the name "gradient"? Because d f d f df\boldsymbol{d} fdf describes the first order changes in f f fff in the neighborhood of O 0 O 0 O_(0)\mathscr{\mathscr { O }}_{0}O0 :
(2.15) f ( P ) = f ( P 0 ) + d f , P P 0 + ( nonlinear terms ) . (2.15) f ( P ) = f P 0 + d f , P P 0 + (  nonlinear terms  ) . {:(2.15)f(P)=f(P_(0))+(:df,P-P_(0):)+(" nonlinear terms ").:}\begin{equation*} f(\mathscr{P})=f\left(\mathscr{P}_{0}\right)+\left\langle\boldsymbol{d} f, \mathscr{P}-\mathscr{P}_{0}\right\rangle+(\text { nonlinear terms }) . \tag{2.15} \end{equation*}(2.15)f(P)=f(P0)+df,PP0+( nonlinear terms ).
[Compare the fundamental idea of "derivative" of something as "best linear approximation to that something at a point"-an idea that works even for functions whose values and arguments are infinite dimensional vectors! See, e.g., Dieudonné (1960).]
Take any vector v v v\boldsymbol{v}v; construct the curve P ( λ ) P ( λ ) P(lambda)\mathscr{P}(\lambda)P(λ) defined by P ( λ ) P 0 = λ v P ( λ ) P 0 = λ v P(lambda)-P_(0)=lambda v\mathscr{P}(\lambda)-\mathscr{P}_{0}=\lambda \boldsymbol{v}P(λ)P0=λv; and differentiate the function f f fff along this curve:
(2.16a) v f = ( d / d λ ) λ = 0 f [ P ( λ ) ] = ( d f / d λ ) P 0 (2.16a) v f = ( d / d λ ) λ = 0 f [ P ( λ ) ] = ( d f / d λ ) P 0 {:(2.16a)del_(v)f=(d//d lambda)_(lambda=0)f[P(lambda)]=(df//d lambda)_(P_(0)):}\begin{equation*} \partial_{\mathbf{v}} f=(d / d \lambda)_{\lambda=0} f[\mathscr{P}(\lambda)]=(d f / d \lambda)_{\mathscr{P}_{0}} \tag{2.16a} \end{equation*}(2.16a)vf=(d/dλ)λ=0f[P(λ)]=(df/dλ)P0
The "differential operator,"
(2.16b) v = ( d / d λ ) at λ = 0 , along curve φ ( λ ) ρ 0 = λ v , , (2.16b) v = ( d / d λ ) at  λ = 0 ,  along curve  φ ( λ ) ρ 0 = λ v , {:(2.16b)del_(v)=(d//d lambda)_("at "lambda=0," along curve "varphi(lambda)-rho_(0)=lambda v,)^(", "):}\begin{equation*} \partial_{\boldsymbol{v}}=(d / d \lambda)_{\text {at } \lambda=0, \text { along curve } \varphi(\lambda)-\rho_{0}=\lambda \boldsymbol{v}, ~}^{\text {, }} \tag{2.16b} \end{equation*}(2.16b)v=(d/dλ)at λ=0, along curve φ(λ)ρ0=λv, 
Gradient of a function as a 1-form
Directional derivative operator defined
Basis 1-forms
which does this differentiating, is called the "directional derivative operator along the vector v v v\boldsymbol{v}v." The directional derivative v f v f del_(v)f\partial_{\mathbf{v}} fvf and the gradient d f d f df\boldsymbol{d} fdf are intimately related, as one sees by applying v v del_(v)\partial_{v}v to equation (2.15) and evaluating the result at the point P 0 P 0 P_(0)\mathscr{\mathscr { P }}_{0}P0 :
(2.17) v f = d f , d P / d λ = d f , v . (2.17) v f = d f , d P / d λ = d f , v . {:(2.17)del_(v)f=(:df","dP//d lambda:)=(:df","v:).:}\begin{equation*} \partial_{\boldsymbol{v}} f=\langle\boldsymbol{d} f, d \mathscr{P} / d \lambda\rangle=\langle\boldsymbol{d} f, \boldsymbol{v}\rangle . \tag{2.17} \end{equation*}(2.17)vf=df,dP/dλ=df,v.
This result, expressed in words, is: d f d f df\boldsymbol{d} fdf is a linear machine for computing the rate of change of f f fff along any desired vector v v v\boldsymbol{v}v. Insert v v v\boldsymbol{v}v into d f d f df\boldsymbol{d} fdf; the output ("number of surfaces pierced; number of bongs of bell") is v f v f del_(v)f\partial_{\mathbf{v}} fvf-which, for sufficiently small v v v\mathbf{v}v, is simply the difference in f f fff between tip and tail of v v v\boldsymbol{v}v.

§2.7. COORDINATE REPRESENTATION OF GEOMETRIC OBJECTS

In flat spacetime, special attention focuses on Lorentz frames. The coordinates x 0 ( P ) x 0 ( P ) x^(0)(P)x^{0}(\mathscr{P})x0(P), x 1 ( P ) , x 2 ( P ) , x 3 ( P ) x 1 ( P ) , x 2 ( P ) , x 3 ( P ) x^(1)(P),x^(2)(P),x^(3)(P)x^{1}(\mathscr{P}), x^{2}(\mathscr{P}), x^{3}(\mathscr{P})x1(P),x2(P),x3(P) of a Lorentz frame are functions; so their gradients can be calculated. Each of the resulting "basis 1 -forms,"
(2.18) ω α = d x α (2.18) ω α = d x α {:(2.18)omega^(alpha)=dx^(alpha):}\begin{equation*} \boldsymbol{\omega}^{\alpha}=\boldsymbol{d} x^{\alpha} \tag{2.18} \end{equation*}(2.18)ωα=dxα
has as its hyperplanes the coordinate surfaces x α = x α = x^(alpha)=x^{\alpha}=xα= const; see Figure 2.8. Consequently the basis vector e α e α e_(alpha)\boldsymbol{e}_{\alpha}eα pierces precisely one surface of the basis 1-form ω α ω α omega^(alpha)\boldsymbol{\omega}^{\alpha}ωα,
Figure 2.8.
The basis vectors and 1 -forms of a particular Lorentz coordinate frame. The basis 1 -forms are so laid out that
ω α , e β = δ α β . ω α , e β = δ α β . (:omega^(alpha),e_(beta):)=delta^(alpha)_(beta).\left\langle\boldsymbol{\omega}^{\alpha}, \boldsymbol{e}_{\beta}\right\rangle=\delta^{\alpha}{ }_{\beta} .ωα,eβ=δαβ.
while the other three basis vectors lie parallel to the surfaces of ω α ω α omega^(alpha)\boldsymbol{\omega}^{\alpha}ωα and thus pierce none:
(2.19) ω α , e β = δ α β . (2.19) ω α , e β = δ α β . {:(2.19)(:omega^(alpha),e_(beta):)=delta^(alpha)_(beta).:}\begin{equation*} \left\langle\boldsymbol{\omega}^{\alpha}, \boldsymbol{e}_{\beta}\right\rangle=\delta^{\alpha}{ }_{\beta} . \tag{2.19} \end{equation*}(2.19)ωα,eβ=δαβ.
(One says that the set of basis 1-forms { ω α } ω α {omega^(alpha)}\left\{\boldsymbol{\omega}^{\alpha}\right\}{ωα} and the set of basis vectors { e β } e β {e_(beta)}\left\{\boldsymbol{e}_{\beta}\right\}{eβ} are the "duals" of each other if they have this property.)
Just as arbitrary vectors can be expanded in terms of the basis e α , v = v α e α e α , v = v α e α e_(alpha),v=v^(alpha)e_(alpha)\boldsymbol{e}_{\alpha}, \boldsymbol{v}=v^{\alpha} \boldsymbol{e}_{\alpha}eα,v=vαeα, so arbitrary 1 -forms can be expanded in terms of ω β ω β omega^(beta)\boldsymbol{\omega}^{\beta}ωβ :
(2.20) σ = σ β ω β . (2.20) σ = σ β ω β . {:(2.20)sigma=sigma_(beta)omega^(beta).:}\begin{equation*} \boldsymbol{\sigma}=\sigma_{\beta} \boldsymbol{\omega}^{\beta} . \tag{2.20} \end{equation*}(2.20)σ=σβωβ.
The expansion coefficients σ β σ β sigma_(beta)\sigma_{\beta}σβ are called "the components of σ σ sigma\boldsymbol{\sigma}σ on the basis ω β ω β omega^(beta)\boldsymbol{\omega}^{\beta}ωβ."
These definitions produce an elegant computational formalism, thus: Calculate how many surfaces of σ σ sigma\boldsymbol{\sigma}σ are pierced by the basis vector e α e α e_(alpha)\boldsymbol{e}_{\alpha}eα; equations (2.19) and (2.20) give the answer:
σ , e α = σ β ω β , e α = σ β ω β , e α = σ β δ β α ; σ , e α = σ β ω β , e α = σ β ω β , e α = σ β δ β α ; (:sigma,e_(alpha):)=(:sigma_(beta)omega^(beta),e_(alpha):)=sigma_(beta)(:omega^(beta),e_(alpha):)=sigma_(beta)delta^(beta)_(alpha);\left\langle\boldsymbol{\sigma}, \boldsymbol{e}_{\alpha}\right\rangle=\left\langle\sigma_{\beta} \boldsymbol{\omega}^{\beta}, \boldsymbol{e}_{\alpha}\right\rangle=\sigma_{\beta}\left\langle\boldsymbol{\omega}^{\beta}, \boldsymbol{e}_{\alpha}\right\rangle=\sigma_{\beta} \delta^{\beta}{ }_{\alpha} ;σ,eα=σβωβ,eα=σβωβ,eα=σβδβα;
i.e.,
(2.21a) σ , e α = σ α . (2.21a) σ , e α = σ α . {:(2.21a)(:sigma,e_(alpha):)=sigma_(alpha).:}\begin{equation*} \left\langle\boldsymbol{\sigma}, \boldsymbol{e}_{\alpha}\right\rangle=\sigma_{\alpha} . \tag{2.21a} \end{equation*}(2.21a)σ,eα=σα.
Similarly, calculate ω α , v ω α , v (:omega^(alpha),v:)\left\langle\boldsymbol{\omega}^{\alpha}, \boldsymbol{v}\right\rangleωα,v for any vector v = e β v β v = e β v β v=e_(beta)v^(beta)\boldsymbol{v}=\boldsymbol{e}_{\beta} v^{\beta}v=eβvβ; the result is
(2.21b) ω α , v = v α . (2.21b) ω α , v = v α . {:(2.21b)(:omega^(alpha),v:)=v^(alpha).:}\begin{equation*} \left\langle\boldsymbol{\omega}^{\alpha}, \boldsymbol{v}\right\rangle=v^{\alpha} . \tag{2.21b} \end{equation*}(2.21b)ωα,v=vα.
Multiply equation (2.21a) by v α v α v^(alpha)v^{\alpha}vα and sum, or multiply (2.21b) by σ α σ α sigma_(alpha)\sigma_{\alpha}σα and sum; the result in either case is
(2.22) σ , v = σ α v α . (2.22) σ , v = σ α v α . {:(2.22)(:sigma","v:)=sigma_(alpha)v^(alpha).:}\begin{equation*} \langle\boldsymbol{\sigma}, \boldsymbol{v}\rangle=\sigma_{\alpha} v^{\alpha} . \tag{2.22} \end{equation*}(2.22)σ,v=σαvα.
This provides a way, using components, to calculate the coordinate-independent value of σ , v σ , v (:sigma,v:)\langle\boldsymbol{\sigma}, \boldsymbol{v}\rangleσ,v.
Each Lorentz frame gives a coordinate-dependent representation of any geometric object or relation: v v v\boldsymbol{v}v is represented by its components v α ; σ v α ; σ v^(alpha);sigmav^{\alpha} ; \boldsymbol{\sigma}vα;σ, by its components σ α σ α sigma_(alpha)\sigma_{\alpha}σα; a point P P P\mathscr{P}P, by its coordinates x α x α x^(alpha)x^{\alpha}xα; the relation σ , v = 17.3 σ , v = 17.3 (:sigma,v:)=17.3\langle\boldsymbol{\sigma}, \boldsymbol{v}\rangle=17.3σ,v=17.3 by σ α v α = 17.3 σ α v α = 17.3 sigma_(alpha)v^(alpha)=17.3\sigma_{\alpha} v^{\alpha}=17.3σαvα=17.3.
To find the coordinate representation of the directional derivative operator v v del_(v)\partial_{v}v, rewrite equation (2.16b) using elementary calculus
v = ( d d λ ) P 0 = ( d x α d λ ) at P 0 along P ( λ ) Q 0 = λ v v α ; see equation ( 2.3 ) ( x α ) ; v = d d λ P 0 = d x α d λ at  P 0  along  P ( λ ) Q 0 = λ v v α ;  see equation  ( 2.3 ) x α ; del_(v)=((d)/(d lambda))_(P_(0))=ubrace(((dx^(alpha))/(d lambda))_("at "P_(0)" along "P(lambda)-Q_(0)=lambda v)ubrace)_(v^(alpha);" see equation "(2.3))((del)/(delx^(alpha)));\partial_{v}=\left(\frac{d}{d \lambda}\right)_{\mathscr{P}_{0}}=\underbrace{\left(\frac{d x^{\alpha}}{d \lambda}\right)_{\text {at } \mathscr{P}_{0} \text { along } \mathscr{P}(\lambda)-\mathscr{Q}_{0}=\lambda v}}_{v^{\alpha} ; \text { see equation }(2.3)}\left(\frac{\partial}{\partial x^{\alpha}}\right) ;v=(ddλ)P0=(dxαdλ)at P0 along P(λ)Q0=λvvα; see equation (2.3)(xα);
the result is
(2.23) v = v α / x α (2.23) v = v α / x α {:(2.23)del_(v)=v^(alpha)del//delx^(alpha):}\begin{equation*} \partial_{\boldsymbol{v}}=v^{\alpha} \partial / \partial x^{\alpha} \tag{2.23} \end{equation*}(2.23)v=vα/xα
Directional derivative in terms of coordinates
In particular, the directional derivative along a basis vector e α e α e_(alpha)\boldsymbol{e}_{\alpha}eα (components [ e α ] β = ω β , e α = δ β α e α β = ω β , e α = δ β α [e_(alpha)]^(beta)=(:omega^(beta),e_(alpha):)=delta^(beta)_(alpha)\left[\boldsymbol{e}_{\alpha}\right]^{\beta}=\left\langle\boldsymbol{\omega}^{\beta}, \boldsymbol{e}_{\alpha}\right\rangle=\delta^{\beta}{ }_{\alpha}[eα]β=ωβ,eα=δβα ) is
(2.24) α e a = / x α . (2.24) α e a = / x α . {:(2.24)del_(alpha)-=del_(e_(a))=del//delx^(alpha).:}\begin{equation*} \partial_{\alpha} \equiv \partial_{\boldsymbol{e}_{\boldsymbol{a}}}=\partial / \partial x^{\alpha} . \tag{2.24} \end{equation*}(2.24)αea=/xα.
This should also be obvious from Figure 2.8.
The components of the gradient 1-form d f d f df\boldsymbol{d} fdf, which are denoted f , α f , α f_(,alpha)f_{, \alpha}f,α
(2.25a) d f = f , α ω α , (2.25a) d f = f , α ω α , {:(2.25a)df=f_(,alpha)omega^(alpha)",":}\begin{equation*} \boldsymbol{d} f=f_{, \alpha} \boldsymbol{\omega}^{\alpha}, \tag{2.25a} \end{equation*}(2.25a)df=f,αωα,
are calculated easily using the above formulas:
f , α = d f , e α [ standard way to calculate components; equation (2.21a)] = α f [ by relation (2.17) between directional derivative and gradient ] = f / x α [ by equation (2.24)]. f , α = d f , e α [  standard way to calculate components; equation (2.21a)]  = α f [  by relation (2.17) between directional derivative and gradient  ] = f / x α [  by equation (2.24)].  {:[f_(,alpha)=(:df,e_(alpha):)[" standard way to calculate components; equation (2.21a)] "],[=del_(alpha)f quad[" by relation (2.17) between directional derivative and gradient "]],[=del f//delx^(alpha)quad[" by equation (2.24)]. "]:}\begin{aligned} f_{, \alpha} & =\left\langle\boldsymbol{d} f, \boldsymbol{e}_{\alpha}\right\rangle[\text { standard way to calculate components; equation (2.21a)] } \\ & =\partial_{\alpha} f \quad[\text { by relation (2.17) between directional derivative and gradient }] \\ & =\partial f / \partial x^{\alpha} \quad[\text { by equation (2.24)]. } \end{aligned}f,α=df,eα[ standard way to calculate components; equation (2.21a)] =αf[ by relation (2.17) between directional derivative and gradient ]=f/xα[ by equation (2.24)]. 
Thus, in agreement with the elementary calculus idea of gradient, the components of d f d f df\boldsymbol{d} fdf are just the partial derivatives along the coordinate axes:
(2.25b) f , α = f / x α ; i.e., d f = ( f / x α ) d x α . (2.25b) f , α = f / x α ;  i.e.,  d f = f / x α d x α . {:(2.25b)f_(,alpha)=del f//delx^(alpha);quad" i.e., "df=(del f//delx^(alpha))dx^(alpha).:}\begin{equation*} f_{, \alpha}=\partial f / \partial x^{\alpha} ; \quad \text { i.e., } \boldsymbol{d} f=\left(\partial f / \partial x^{\alpha}\right) \boldsymbol{d} x^{\alpha} . \tag{2.25b} \end{equation*}(2.25b)f,α=f/xα; i.e., df=(f/xα)dxα.
(Recall: ω α = d x α ω α = d x α omega^(alpha)=dx^(alpha)\boldsymbol{\omega}^{\alpha}=\boldsymbol{d} x^{\alpha}ωα=dxα.) The formula d f = ( f / x α ) d x α d f = f / x α d x α df=(del f//delx^(alpha))dx^(alpha)\boldsymbol{d} f=\left(\partial f / \partial x^{\alpha}\right) \boldsymbol{d} x^{\alpha}df=(f/xα)dxα suggests, correctly, that d f d f df\boldsymbol{d} fdf is a rigorous version of the "differential" of elementary calculus; see Box 2.3.
Other important coordinate representations for geometric relations are explored in the following exercises.

EXERCISES

Derive the following computationally useful formulas:

Exercise 2.2. LOWERING INDEX TO GET THE 1-FORM CORRESPONDING TO A VECTOR

The components u α u α u_(alpha)u_{\alpha}uα of the 1 -form u ~ u ~ widetilde(u)\widetilde{\boldsymbol{u}}u~ that corresponds to a vector u u u\boldsymbol{u}u can be obtained by "lowering an index" with the metric coefficients η α β η α β eta_(alpha beta)\eta_{\alpha \beta}ηαβ :
(2.26a) u α = η α β u β ; i.e., u 0 = u 0 , u k = u k . (2.26a) u α = η α β u β ;  i.e.,  u 0 = u 0 , u k = u k . {:(2.26a)u_(alpha)=eta_(alpha beta)u^(beta);" i.e., "quadu_(0)=-u^(0)","u_(k)=u^(k).:}\begin{equation*} u_{\alpha}=\eta_{\alpha \beta} u^{\beta} ; \text { i.e., } \quad u_{0}=-u^{0}, u_{k}=u^{k} . \tag{2.26a} \end{equation*}(2.26a)uα=ηαβuβ; i.e., u0=u0,uk=uk.

Exercise 2.3. RAISING INDEX TO RECOVER THE VECTOR

One can return to the components of u u u\boldsymbol{u}u by raising indices,
(2.26b) u α = η α β u β (2.26b) u α = η α β u β {:(2.26b)u^(alpha)=eta^(alpha beta)u_(beta):}\begin{equation*} u^{\alpha}=\eta^{\alpha \beta} u_{\beta} \tag{2.26b} \end{equation*}(2.26b)uα=ηαβuβ
the matrix η α β η α β ||eta^(alpha beta)||\left\|\eta^{\alpha \beta}\right\|ηαβ is defined as the inverse of η α β η α β ||eta_(alpha beta)||\left\|\eta_{\alpha \beta}\right\|ηαβ, and happens to equal η α β η α β ||eta_(alpha beta)||\left\|\eta_{\alpha \beta}\right\|ηαβ :
(2.27) η α β η β γ δ α γ ; η α β = η α β for all α , β (2.27) η α β η β γ δ α γ ; η α β = η α β  for all  α , β {:(2.27)eta^(alpha beta)eta_(beta gamma)-=delta^(alpha)_(gamma);quadeta^(alpha beta)=eta_(alpha beta)" for all "alpha","beta:}\begin{equation*} \eta^{\alpha \beta} \eta_{\beta \gamma} \equiv \delta^{\alpha}{ }_{\gamma} ; \quad \eta^{\alpha \beta}=\eta_{\alpha \beta} \text { for all } \alpha, \beta \tag{2.27} \end{equation*}(2.27)ηαβηβγδαγ;ηαβ=ηαβ for all α,β

Exercise 2.4. VARIED ROUTES TO THE SCALAR PRODUCT

The scalar product of u u u\boldsymbol{u}u with v v v\boldsymbol{v}v can be calculated in any of the following ways:
(2.28) u v = g ( u , v ) = u α v β η α β = u α v α = u α v β η α β (2.28) u v = g ( u , v ) = u α v β η α β = u α v α = u α v β η α β {:(2.28)u*v=g(u","v)=u^(alpha)v^(beta)eta_(alpha beta)=u^(alpha)v_(alpha)=u_(alpha)v_(beta)eta^(alpha beta):}\begin{equation*} \boldsymbol{u} \cdot \boldsymbol{v}=\boldsymbol{g}(\boldsymbol{u}, \boldsymbol{v})=u^{\alpha} v^{\beta} \eta_{\alpha \beta}=u^{\alpha} v_{\alpha}=u_{\alpha} v_{\beta} \eta^{\alpha \beta} \tag{2.28} \end{equation*}(2.28)uv=g(u,v)=uαvβηαβ=uαvα=uαvβηαβ

Box 2.3 DIFFERENTIALS

The "exterior derivative" or "gradient" d f d f df\boldsymbol{d} fdf of a function f f fff is a more rigorous version of the elementary concept of "differential."
In elementary textbooks, one is presented with the differential d f d f dfd fdf as representing "an infinitesimal change in the function f ( P ) f ( P ) f(P)f(\mathscr{P})f(P) " associated with some infinitesimal displacement of the point P P P\mathscr{P}P; but one will recall that the displacement of P P P\mathscr{P}P is left arbitrary, albeit infinitesimal. Thus d f d f dfd fdf represents a change in f f fff in some unspecified direction.
But this is precisely what the exterior derivative d f d f df\boldsymbol{d} fdf represents. Choose a particular, infinitesimally long displacement v v v\boldsymbol{v}v of the point P P P\mathscr{P}P. Let the dis-
placement vector v v v\boldsymbol{v}v pierce d f d f df\boldsymbol{d} fdf to give the number d f , v = v f d f , v = v f (:df,v:)=del_(v)f\langle\boldsymbol{d} f, \boldsymbol{v}\rangle=\partial_{\boldsymbol{v}} fdf,v=vf. That number is the change of f f fff in going from the tail of v v v\boldsymbol{v}v to its tip. Thus d f d f df\boldsymbol{d} fdf, before it has been pierced to give a number, represents the change of f f fff in an unspecified direction. The act of piercing d f d f df\boldsymbol{d} fdf with v v v\boldsymbol{v}v is the act of making explicit the direction in which the change is to be measured. The only failing of the textbook presentation, then, was its suggestion that d f d f df\boldsymbol{d} fdf was a scalar or a number; the explicit recognition of the need for specifying a direction v v v\boldsymbol{v}v to reduce d f d f df\boldsymbol{d} fdf to a number d f , v d f , v (:df,v:)\langle\boldsymbol{d} f, \boldsymbol{v}\rangledf,v shows that in fact d f d f df\boldsymbol{d} fdf is a 1-form, the gradient of f f fff.

§2.8. THE CENTRIFUGE AND THE PHOTON

Vectors, metric, 1-forms, functions, gradients, directional derivatives: all these geometric objects and more are used in flat spacetime to represent physical quantities; and all the laws of physics must be expressible in terms of such geometric objects.
As an example, consider a high-precision redshift experiment that uses the Mössbauer effect (Figure 2.9). The emitter and the absorber of photons are attached to
Figure 2.9.
The centrifuge and the photon.
the rim of a centrifuge at points separated by an angle α α alpha\alphaα, as measured in the inertial laboratory. The emitter and absorber are at radius r r rrr as measured in the laboratory, and the centrifuge rotates with angular velocity ω ω omega\omegaω. Problem: What is the redshift measured,
z = ( λ absorbed λ emitted ) / λ emitted z = λ absorbed  λ emitted  / λ emitted  z=(lambda_("absorbed ")-lambda_("emitted "))//lambda_("emitted ")z=\left(\lambda_{\text {absorbed }}-\lambda_{\text {emitted }}\right) / \lambda_{\text {emitted }}z=(λabsorbed λemitted )/λemitted 
in terms of ω , r ω , r omega,r\omega, rω,r, and α α alpha\alphaα ?
Solution: Let u e u e u_(e)\boldsymbol{u}_{e}ue be the 4-velocity of the emitter at the event of emission of a given photon; let u a u a u_(a)\boldsymbol{u}_{a}ua be the 4 -velocity of the absorber at the event of absorption; and let p p p\boldsymbol{p}p be the 4 -momentum of the photon. All three quantities are vectors defined without reference to coordinates. Equally coordinate-free are the photon energies E e E e E_(e)E_{e}Ee and E a E a E_(a)E_{a}Ea measured by emitter and absorber. No coordinates are needed to describe the fact that a specific emitter emitting a specific photon attributes to it the energy E e E e E_(e)E_{e}Ee; and no coordinates are required in the geometric formula
(2.29) E e = p u e (2.29) E e = p u e {:(2.29)E_(e)=-p*u_(e):}\begin{equation*} E_{e}=-\boldsymbol{p} \cdot \boldsymbol{u}_{e} \tag{2.29} \end{equation*}(2.29)Ee=pue
for E e E e E_(e)E_{e}Ee. [That this formula works can be readily verified by recalling that, in the emitter's frame, u e 0 = 1 u e 0 = 1 u_(e)^(0)=1u_{e}{ }^{0}=1ue0=1 and u e j = 0 u e j = 0 u_(e)^(j)=0u_{e}{ }^{j}=0uej=0; so
E e = p α u e α = p 0 = + p 0 E e = p α u e α = p 0 = + p 0 E_(e)=-p_(alpha)u_(e)^(alpha)=-p_(0)=+p^(0)E_{e}=-p_{\alpha} u_{e}^{\alpha}=-p_{0}=+p^{0}Ee=pαueα=p0=+p0
in accordance with the identification "(time component of 4-momentum) = ( = ( =(=(=( energy."] Analogous to equation (2.29) is the purely geometric formula
E a = p u a E a = p u a E_(a)=-p*u_(a)E_{a}=-\boldsymbol{p} \cdot \boldsymbol{u}_{a}Ea=pua
for the absorbed energy.
The ratio of absorbed wavelength to emitted wavelength is the inverse of the energy ratio (since E = h ν = h c / λ E = h ν = h c / λ E=h nu=hc//lambdaE=h \nu=h c / \lambdaE=hν=hc/λ ):
λ a λ e = E e E a = p u e p u a . λ a λ e = E e E a = p u e p u a . (lambda_(a))/(lambda_(e))=(E_(e))/(E_(a))=(-p*u_(e))/(-p*u_(a)).\frac{\lambda_{a}}{\lambda_{e}}=\frac{E_{e}}{E_{a}}=\frac{-\boldsymbol{p} \cdot \boldsymbol{u}_{e}}{-\boldsymbol{p} \cdot \boldsymbol{u}_{a}} .λaλe=EeEa=puepua.
This ratio is most readily calculated in the inertial laboratory frame
(2.30) λ a λ e = p 0 u e 0 p j u e j p 0 u a 0 p j u a j p 0 u e 0 p u e p 0 u a 0 p u a . (2.30) λ a λ e = p 0 u e 0 p j u e j p 0 u a 0 p j u a j p 0 u e 0 p u e p 0 u a 0 p u a . {:(2.30)(lambda_(a))/(lambda_(e))=(p^(0)u_(e)^(0)-p^(j)u_(e)^(j))/(p^(0)u_(a)^(0)-p^(j)u_(a)^(j))-=(p^(0)u_(e)^(0)-p*u_(e))/(p^(0)u_(a)^(0)-p*u_(a)).:}\begin{equation*} \frac{\lambda_{a}}{\lambda_{e}}=\frac{p^{0} u_{e}{ }^{0}-p^{j} u_{e}{ }^{j}}{p^{0} u_{a}{ }^{0}-p^{j} u_{a}{ }^{j}} \equiv \frac{p^{0} u_{e}{ }^{0}-\boldsymbol{p} \cdot \boldsymbol{u}_{e}}{p^{0} u_{a}{ }^{0}-\boldsymbol{p} \cdot u_{a}} . \tag{2.30} \end{equation*}(2.30)λaλe=p0ue0pjuejp0ua0pjuajp0ue0puep0ua0pua.
(Here and throughout we use boldface Latin letters for three-dimensional vectors in a given Lorentz frame; and we use the usual notation and formalism of threedimensional, Euclidean vector analysis to manipulate them.) Because the magnitude of the ordinary velocity of the rim of the centrifuge, v = ω r v = ω r v=omega rv=\omega rv=ωr, is unchanging in time, u e 0 u e 0 u_(e)^(0)u_{e}{ }^{0}ue0 and u a 0 u a 0 u_(a)^(0)u_{a}{ }^{0}ua0 are equal, and the magnitudes-but not the directions-of u e u e u_(e)\boldsymbol{u}_{e}ue and u a u a u_(a)\boldsymbol{u}_{a}ua are equal:
u e 0 = u a 0 = ( 1 v 2 ) 1 / 2 , | u e | = | u a | = v / ( 1 v 2 ) 1 / 2 . u e 0 = u a 0 = 1 v 2 1 / 2 , u e = u a = v / 1 v 2 1 / 2 . u_(e)^(0)=u_(a)^(0)=(1-v^(2))^(-1//2),|u_(e)|=|u_(a)|=v//(1-v^(2))^(1//2).u_{e}{ }^{0}=u_{a}^{0}=\left(1-v^{2}\right)^{-1 / 2},\left|\boldsymbol{u}_{e}\right|=\left|\boldsymbol{u}_{a}\right|=v /\left(1-v^{2}\right)^{1 / 2} .ue0=ua0=(1v2)1/2,|ue|=|ua|=v/(1v2)1/2.
From the geometry of Figure 2.9 , one sees that u e u e u_(e)\boldsymbol{u}_{e}ue makes the same angle with p p p\boldsymbol{p}p as does u a u a u_(a)\boldsymbol{u}_{a}ua. Consequently, p u e = p u a p u e = p u a p*u_(e)=p*u_(a)\boldsymbol{p} \cdot \boldsymbol{u}_{e}=\boldsymbol{p} \cdot \boldsymbol{u}_{a}pue=pua, and λ absorbed / λ emitted = 1 λ absorbed  / λ emitted  = 1 lambda_("absorbed ")//lambda_("emitted ")=1\lambda_{\text {absorbed }} / \lambda_{\text {emitted }}=1λabsorbed /λemitted =1. There is no redshift!
Notice that this solution made no reference whatsoever to Lorentz transforma-tions-they have not even been discussed yet in this book! The power of the geometric, coordinate-free viewpoint is evident!
One must have a variety of coordinate-free contacts between theory and experiment in order to use the geometric viewpoint. One such contact is the equation E = p u E = p u E=-p*uE=-\boldsymbol{p} \cdot \boldsymbol{u}E=pu for the energy of a photon with 4 -momentum p p p\boldsymbol{p}p, as measured by an observer with 4 -velocity u u u\boldsymbol{u}u. Verify the following other points of contact.

Exercise 2.5. ENERGY AND VELOCITY FROM 4-MOMENTUM

A particle of rest mass m m mmm and 4-momentum p p p\boldsymbol{p}p is examined by an observer with 4 -velocity
u u u\boldsymbol{u}u. Show that just as (a) the energy he measures is
(2.31) E = p u (2.31) E = p u {:(2.31)E=-p*u:}\begin{equation*} E=-\boldsymbol{p} \cdot \boldsymbol{u} \tag{2.31} \end{equation*}(2.31)E=pu
so (b) the rest mass he attributes to the particle is
(2.32) m 2 = p 2 (2.32) m 2 = p 2 {:(2.32)m^(2)=-p^(2):}\begin{equation*} m^{2}=-\boldsymbol{p}^{2} \tag{2.32} \end{equation*}(2.32)m2=p2
(c) the momentum he measures has magnitude
(2.33) | p | = [ ( p u ) 2 + ( p p ) ] 1 / 2 (2.33) | p | = ( p u ) 2 + ( p p ) 1 / 2 {:(2.33)|p|=[(p*u)^(2)+(p*p)]^(1//2):}\begin{equation*} |\boldsymbol{p}|=\left[(\boldsymbol{p} \cdot \boldsymbol{u})^{2}+(\boldsymbol{p} \cdot \boldsymbol{p})\right]^{1 / 2} \tag{2.33} \end{equation*}(2.33)|p|=[(pu)2+(pp)]1/2
(d) the ordinary velocity v v vvv he measures has magnitude
(2.34) | v | = | p | E (2.34) | v | = | p | E {:(2.34)|v|=(|p|)/(E):}\begin{equation*} |v|=\frac{|p|}{E} \tag{2.34} \end{equation*}(2.34)|v|=|p|E
where | p | | p | |p||\boldsymbol{p}||p| and E E EEE are as given above; and (e) the 4 -vector v v v\boldsymbol{v}v, whose components in the observer's Lorentz frame are
v 0 = 0 , v j = ( d x j / d t ) for particle = ordinary velocity v 0 = 0 , v j = d x j / d t for particle  =  ordinary velocity  v^(0)=0,quadv^(j)=(dx^(j)//dt)_("for particle ")=" ordinary velocity "v^{0}=0, \quad v^{j}=\left(d x^{j} / d t\right)_{\text {for particle }}=\text { ordinary velocity }v0=0,vj=(dxj/dt)for particle = ordinary velocity 
is given by
(2.35) v = p + ( p u ) u p u (2.35) v = p + ( p u ) u p u {:(2.35)v=(p+(p*u)u)/(-p*u):}\begin{equation*} v=\frac{p+(p \cdot u) u}{-p \cdot u} \tag{2.35} \end{equation*}(2.35)v=p+(pu)upu

Exercise 2.6. TEMPERATURE GRADIENT

To each event Q Q Q\mathscr{Q}Q inside the sun one attributes a temperature T ( Q ) T ( Q ) T(Q)T(\mathcal{Q})T(Q), the temperature measured by a thermometer at rest in the hot gas there. Then T ( Q ) T ( Q ) T(Q)T(\mathscr{Q})T(Q) is a function; no coordinates are required for its definition and discussion. A cosmic ray from outer space flies through the sun with 4 -velocity u u u\boldsymbol{u}u. Show that, as measured by the cosmic ray's clock, the time derivative of temperature in its vicinity is
(2.36) d T / d τ = u T = d T , u (2.36) d T / d τ = u T = d T , u {:(2.36)dT//d tau=del_(u)T=(:dT","u:):}\begin{equation*} d T / d \tau=\partial_{\boldsymbol{u}} T=\langle\boldsymbol{d} T, \boldsymbol{u}\rangle \tag{2.36} \end{equation*}(2.36)dT/dτ=uT=dT,u
In a local Lorentz frame inside the sun, this equation can be written
(2.37) d T d τ = u α T x α = 1 1 v 2 T t + v j 1 v 2 T x j . (2.37) d T d τ = u α T x α = 1 1 v 2 T t + v j 1 v 2 T x j . {:(2.37)(dT)/(d tau)=u^(alpha)(del T)/(delx^(alpha))=(1)/(sqrt(1-v^(2)))(del T)/(del t)+(v^(j))/(sqrt(1-v^(2)))(del T)/(delx^(j)).:}\begin{equation*} \frac{d T}{d \tau}=u^{\alpha} \frac{\partial T}{\partial x^{\alpha}}=\frac{1}{\sqrt{1-v^{2}}} \frac{\partial T}{\partial t}+\frac{v^{j}}{\sqrt{1-v^{2}}} \frac{\partial T}{\partial x^{j}} . \tag{2.37} \end{equation*}(2.37)dTdτ=uαTxα=11v2Tt+vj1v2Txj.
Why is this result reasonable?

§2.9. LORENTZ TRANSFORMATIONS

To simplify computations, one often works with the components of vectors and 1 -forms, rather than with coordinate-free language. Such component manipulations sometimes involve transformations from one Lorentz frame to another. The reader is already familiar with such Lorentz transformations; but the short review in Box 2.4 will refresh his memory and acquaint him with the notation used in this book.
The key entities in the Lorentz transformation are the matrices Λ α β Λ α β ||Lambda^(alpha^('))_(beta)||\left\|\Lambda^{\alpha^{\prime}}{ }_{\beta}\right\|Λαβ and Λ β α Λ β α ||Lambda^(beta)_(alpha^('))||\left\|\Lambda^{\beta}{ }_{\alpha^{\prime}}\right\|Λβα; the first transforms coordinates from an unprimed frame to a primed frame, while the second goes from primed to unprimed
(2.38) x α = Λ α β x β , x β = Λ β α x α . (2.38) x α = Λ α β x β , x β = Λ β α x α . {:(2.38)x^(alpha^('))=Lambda^(alpha^('))_(beta)x^(beta)","quadx^(beta)=Lambda^(beta)_(alpha^('))x^(alpha^(')).:}\begin{equation*} x^{\alpha^{\prime}}=\Lambda^{\alpha^{\prime}}{ }_{\beta} x^{\beta}, \quad x^{\beta}=\Lambda^{\beta}{ }_{\alpha^{\prime}} x^{\alpha^{\prime}} . \tag{2.38} \end{equation*}(2.38)xα=Λαβxβ,xβ=Λβαxα.
Since they go in opposite directions, each of the two matrices must be the inverse of the other:
(2.39) Λ α Λ β γ = δ α γ ; Λ β α Λ α γ = δ β γ . (2.39) Λ α Λ β γ = δ α γ ; Λ β α Λ α γ = δ β γ . {:(2.39)Lambda^(alpha^('))Lambda^(beta)_(gamma)^(')=delta^(alpha^('))_(gamma^('));quadLambda^(beta)_(alpha^('))Lambda^(alpha^('))_(gamma)=delta^(beta)_(gamma).:}\begin{equation*} \Lambda^{\alpha^{\prime}} \Lambda^{\beta}{ }_{\gamma}{ }^{\prime}=\delta^{\alpha^{\prime}}{ }_{\gamma^{\prime}} ; \quad \Lambda^{\beta}{ }_{\alpha^{\prime}} \Lambda^{\alpha^{\prime}}{ }_{\gamma}=\delta^{\beta}{ }_{\gamma} . \tag{2.39} \end{equation*}(2.39)ΛαΛβγ=δαγ;ΛβαΛαγ=δβγ.
From the coordinate-independent nature of 4-velocity, u = ( d x α / d τ ) e α u = d x α / d τ e α u=(dx^(alpha)//d tau)e_(alpha)\boldsymbol{u}=\left(d x^{\alpha} / d \tau\right) \boldsymbol{e}_{\alpha}u=(dxα/dτ)eα, one readily derives the expressions
(2.40) e α = e β Λ β α , e β = e α Λ α β (2.40) e α = e β Λ β α , e β = e α Λ α β {:(2.40)e_(alpha^('))=e_(beta)Lambda^(beta)_(alpha^('))","quade_(beta)=e_(alpha^('))Lambda^(alpha^('))_(beta):}\begin{equation*} \boldsymbol{e}_{\alpha^{\prime}}=\boldsymbol{e}_{\beta} \Lambda^{\beta}{ }_{\alpha^{\prime}}, \quad \boldsymbol{e}_{\beta}=\boldsymbol{e}_{\alpha^{\prime}} \Lambda^{\alpha^{\prime}}{ }_{\beta} \tag{2.40} \end{equation*}(2.40)eα=eβΛβα,eβ=eαΛαβ
for the basis vectors of one frame in terms of those of the other; and from other geometric equations, such as
v = e α v α = e β v β , σ , v = σ α v α = σ β v β , σ = σ α ω α = σ β ω β , v = e α v α = e β v β , σ , v = σ α v α = σ β v β , σ = σ α ω α = σ β ω β , {:[v=e_(alpha)v^(alpha)=e_(beta^('))*v^(beta^('))","],[(:sigma","v:)=sigma_(alpha)v^(alpha)=sigma_(beta^('))*v^(beta^('))","],[sigma=sigma_(alpha)omega^(alpha)=sigma_(beta^('))*omega^(beta^('))","]:}\begin{aligned} \boldsymbol{v} & =\boldsymbol{e}_{\alpha} v^{\alpha}=\boldsymbol{e}_{\beta^{\prime}} \cdot v^{\beta^{\prime}}, \\ \langle\boldsymbol{\sigma}, \boldsymbol{v}\rangle & =\sigma_{\alpha} v^{\alpha}=\sigma_{\beta^{\prime}} \cdot v^{\beta^{\prime}}, \\ \boldsymbol{\sigma} & =\sigma_{\alpha} \boldsymbol{\omega}^{\alpha}=\sigma_{\beta^{\prime}} \cdot \boldsymbol{\omega}^{\beta^{\prime}}, \end{aligned}v=eαvα=eβvβ,σ,v=σαvα=σβvβ,σ=σαωα=σβωβ,
one derives transformation laws
(2.41) ω α = Λ α ω β , ω β = Λ β α ω α ; (2.42) v α = Λ α v β , v β = Λ β α v α ; (2.43) σ α = σ β Λ β α , σ β = σ α Λ α β . (2.41) ω α = Λ α ω β , ω β = Λ β α ω α ; (2.42) v α = Λ α v β , v β = Λ β α v α ; (2.43) σ α = σ β Λ β α , σ β = σ α Λ α β . {:[(2.41)omega^(alpha^('))=Lambda^(alpha^('))omega^(beta)","omega^(beta)=Lambda^(beta)_(alpha^('))omega^(alpha^('));],[(2.42)v^(alpha^('))=Lambda^(alpha^('))v^(beta)","v^(beta)=Lambda^(beta)_(alpha^('))v^(alpha^('));],[(2.43)sigma_(alpha^('))=sigma_(beta)Lambda^(beta)_(alpha^('))","sigma_(beta)=sigma_(alpha^('))Lambda^(alpha^('))_(beta).]:}\begin{align*} \boldsymbol{\omega}^{\alpha^{\prime}}=\Lambda^{\alpha^{\prime}} \boldsymbol{\omega}^{\beta}, & \boldsymbol{\omega}^{\beta}=\Lambda^{\beta}{ }_{\alpha^{\prime}} \boldsymbol{\omega}^{\alpha^{\prime}} ; \tag{2.41}\\ v^{\alpha^{\prime}}=\Lambda^{\alpha^{\prime}} v^{\beta}, & v^{\beta}=\Lambda^{\beta}{ }_{\alpha^{\prime}} v^{\alpha^{\prime}} ; \tag{2.42}\\ \sigma_{\alpha^{\prime}}=\sigma_{\beta} \Lambda^{\beta}{ }_{\alpha^{\prime}}, & \sigma_{\beta}=\sigma_{\alpha^{\prime}} \Lambda^{\alpha^{\prime}}{ }_{\beta} . \tag{2.43} \end{align*}(2.41)ωα=Λαωβ,ωβ=Λβαωα;(2.42)vα=Λαvβ,vβ=Λβαvα;(2.43)σα=σβΛβα,σβ=σαΛαβ.
One need never memorize the index positions in these transformation laws. One need only line the indices up so that (1) free indices on each side of the equation are in the same position; and (2) summed indices appear once up and once down. Then all will be correct! (Note: the indices on Λ Λ Lambda\LambdaΛ always run "northwest to southeast.")

Box 2.4 LORENTZ TRANSFORMATIONS

Rotation of Frame of Reference by Angle θ θ theta\boldsymbol{\theta}θ in x y x y x-yx-yxy Plane

Slope s = tan θ ; sin θ = s ( 1 + s 2 ) 1 / 2 : cos θ = 1 ( 1 + s 2 ) 1 / 2  Slope  s = tan θ ; sin θ = s 1 + s 2 1 / 2 : cos θ = 1 1 + s 2 1 / 2 " Slope "s=tan theta;quad sin theta=(s)/((1+s^(2))^(1//2)):quad cos theta=(1)/((1+s^(2))^(1//2))\text { Slope } s=\tan \theta ; \quad \sin \theta=\frac{s}{\left(1+s^{2}\right)^{1 / 2}}: \quad \cos \theta=\frac{1}{\left(1+s^{2}\right)^{1 / 2}} Slope s=tanθ;sinθ=s(1+s2)1/2:cosθ=1(1+s2)1/2
t = t ¯ t = t ¯ t= bar(t)t=\bar{t}t=t¯
x = x ¯ cos θ y ¯ sin θ x = x ¯ cos θ y ¯ sin θ x= bar(x)cos theta- bar(y)sin thetax=\bar{x} \cos \theta-\bar{y} \sin \thetax=x¯cosθy¯sinθ
y = x ¯ sin θ + y ¯ cos θ y = x ¯ sin θ + y ¯ cos θ y=ubrace(( bar(x))sin thetaubrace)+ bar(y)cos thetay=\underbrace{\bar{x} \sin \theta}+\bar{y} \cos \thetay=x¯sinθ+y¯cosθ
z = z ¯ z = z ¯ z= bar(z)z=\bar{z}z=z¯
t ¯ = t x ¯ = x cos θ + y sin θ y ¯ = x sin θ + y cos θ z ¯ = z t ¯ = t x ¯ = x cos θ + y sin θ y ¯ = x sin θ + y cos θ z ¯ = z {:[ bar(t)=t],[ bar(x)=x cos theta+y sin theta],[ bar(y)=-x sin theta+y cos theta],[ bar(z)=z]:}\begin{aligned} \bar{t} & =t \\ \bar{x} & =x \cos \theta+y \sin \theta \\ \bar{y} & =-x \sin \theta+y \cos \theta \\ \bar{z} & =z \end{aligned}t¯=tx¯=xcosθ+ysinθy¯=xsinθ+ycosθz¯=z
All signs follow from sign of this term. Positive by inspection of point P P P\mathscr{P}P.

Combination of Two Such Rotations

s = s 1 + s 2 1 s 1 s 2 or θ = θ 1 + θ 2 s = s 1 + s 2 1 s 1 s 2  or  θ = θ 1 + θ 2 s=(s_(1)+s_(2))/(1-s_(1)s_(2))quad" or "quad theta=theta_(1)+theta_(2)s=\frac{s_{1}+s_{2}}{1-s_{1} s_{2}} \quad \text { or } \quad \theta=\theta_{1}+\theta_{2}s=s1+s21s1s2 or θ=θ1+θ2

Boost of Frame of Reference by Velocity Parameter α α alpha\alphaα in z t z t z-tz-tzt Plane

Velocity β = tanh α ; sinh α = β ( 1 β 2 ) 1 / 2 ; cosh α = 1 ( 1 β 2 ) 1 / 2 = " γ β = tanh α ; sinh α = β 1 β 2 1 / 2 ; cosh α = 1 1 β 2 1 / 2 = " γ beta=tanh alpha;quad sinh alpha=(beta)/((1-beta^(2))^(1//2));quad cosh alpha=(1)/((1-beta^(2))^(1//2))="gamma\beta=\tanh \alpha ; \quad \sinh \alpha=\frac{\beta}{\left(1-\beta^{2}\right)^{1 / 2}} ; \quad \cosh \alpha=\frac{1}{\left(1-\beta^{2}\right)^{1 / 2}}=" \gammaβ=tanhα;sinhα=β(1β2)1/2;coshα=1(1β2)1/2="γ "
tan θ = velocity β tan θ =  velocity  β tan theta=" velocity "beta\tan \theta=\text { velocity } \betatanθ= velocity β
= tanh α = tanh α =tanh alpha=\tanh \alpha=tanhα
t = t ¯ cosh α + z ¯ sinh α t = t ¯ cosh α + z ¯ sinh α t= bar(t)cosh alpha+ bar(z)sinh alphat=\bar{t} \cosh \alpha+\bar{z} \sinh \alphat=t¯coshα+z¯sinhα
x = x ¯ x = x ¯ x= bar(x)x=\bar{x}x=x¯
y = y ¯ y = y ¯ y= bar(y)y=\bar{y}y=y¯
t ¯ = t cosh α z sinh α x ¯ = x y ¯ = y z ¯ = t sinh α + z cosh α t ¯ = t cosh α z sinh α x ¯ = x y ¯ = y z ¯ = t sinh α + z cosh α {:[ bar(t)=t cosh alpha-z sinh alpha],[ bar(x)=x],[ bar(y)=y],[ bar(z)=-t sinh alpha+z cosh alpha]:}\begin{aligned} \bar{t} & =t \cosh \alpha-z \sinh \alpha \\ \bar{x} & =x \\ \bar{y} & =y \\ \bar{z} & =-t \sinh \alpha+z \cosh \alpha \end{aligned}t¯=tcoshαzsinhαx¯=xy¯=yz¯=tsinhα+zcoshα
z = t ¯ sinh α + z ¯ cosh α z = t ¯ sinh α + z ¯ cosh α z=ubrace(( bar(t))sinh alphaubrace)+ bar(z)cosh alphaz=\underbrace{\bar{t} \sinh \alpha}+\bar{z} \cosh \alphaz=t¯sinhα+z¯coshα
All signs follow from sign of this term. Positive because object at rest at z ¯ = 0 z ¯ = 0 bar(z)=0\bar{z}=0z¯=0 in rocket frame moves in direction of increasing z z zzz in lab frame.
Matrix notation: x μ = Λ μ ν x p ¯ , x p ¯ = Λ p ¯ μ x μ x μ = Λ μ ν x p ¯ , x p ¯ = Λ p ¯ μ x μ x^(mu)=Lambda^(mu)_(nu)x^( bar(p)),quadx^( bar(p))=Lambda^( bar(p))_(mu)x^(mu)x^{\mu}=\Lambda^{\mu}{ }_{\nu} x^{\bar{p}}, \quad x^{\bar{p}}=\Lambda^{\bar{p}}{ }_{\mu} x^{\mu}xμ=Λμνxp¯,xp¯=Λp¯μxμ
Λ μ ν = cosh α 0 0 sinh α 0 1 0 0 0 0 1 0 sinh α 0 0 cosh α , Λ v ¯ μ = cosh α 0 0 sinh α 0 1 0 0 0 0 1 0 sinh α 0 0 cosh α Λ μ ν = cosh α 0 0 sinh α 0 1 0 0 0 0 1 0 sinh α 0 0 cosh α , Λ v ¯ μ = cosh α 0 0 sinh α 0 1 0 0 0 0 1 0 sinh α 0 0 cosh α ||Lambda^(mu_(nu))||=||[cosh alpha,0,0,sinh alpha],[0,1,0,0],[0,0,1,0],[sinh alpha,0,0,cosh alpha]||,||Lambda^( bar(v))_(mu)||=||[cosh alpha,0,0,-sinh alpha],[0,1,0,0],[0,0,1,0],[-sinh alpha,0,0,cosh alpha]||\left\|\Lambda^{\mu_{\nu}}\right\|=\left\|\begin{array}{cccc}\cosh \alpha & 0 & 0 & \sinh \alpha \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \sinh \alpha & 0 & 0 & \cosh \alpha\end{array}\right\|,\left\|\Lambda^{\bar{v}}{ }_{\mu}\right\|=\left\|\begin{array}{cccc}\cosh \alpha & 0 & 0 & -\sinh \alpha \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ -\sinh \alpha & 0 & 0 & \cosh \alpha\end{array}\right\|Λμν=coshα00sinhα01000010sinhα00coshα,Λv¯μ=coshα00sinhα01000010sinhα00coshα
Energy-momentum 4-vector Charge density-current 4-vector E = E ¯ cosh α + p z ¯ sinh α ρ = ρ ¯ cosh α + j z ¯ sinh α p x = p x ¯ j x = j x ¯ p y = p y ¯ j y = j y ¯ p z = E ¯ sinh α + p z ¯ cosh α j z = ρ ¯ sinh α + j z ¯ cosh α  Energy-momentum 4-vector   Charge density-current 4-vector  E = E ¯ cosh α + p z ¯ sinh α ρ = ρ ¯ cosh α + j z ¯ sinh α p x = p x ¯ j x = j x ¯ p y = p y ¯ j y = j y ¯ p z = E ¯ sinh α + p z ¯ cosh α j z = ρ ¯ sinh α + j z ¯ cosh α {:[" Energy-momentum 4-vector ",," Charge density-current 4-vector "],[E,= bar(E)cosh alpha+p^( bar(z))sinh alpha,rho,= bar(rho)cosh alpha+j^( bar(z))sinh alpha],[p^(x),=p^( bar(x)),,j^(x)=j^( bar(x))],[p^(y),=p^( bar(y)),j^(y),=j^( bar(y))],[p^(z),= bar(E)sinh alpha+p^( bar(z))cosh alpha,j^(z),= bar(rho)sinh alpha+j^( bar(z))cosh alpha]:}\begin{array}{rlrl} \text { Energy-momentum 4-vector } & & \text { Charge density-current 4-vector } \\ E & =\bar{E} \cosh \alpha+p^{\bar{z}} \sinh \alpha & \rho & =\bar{\rho} \cosh \alpha+j^{\bar{z}} \sinh \alpha \\ p^{x} & =p^{\bar{x}} & & j^{x}=j^{\bar{x}} \\ p^{y} & =p^{\bar{y}} & j^{y} & =j^{\bar{y}} \\ p^{z} & =\bar{E} \sinh \alpha+p^{\bar{z}} \cosh \alpha & j^{z} & =\bar{\rho} \sinh \alpha+j^{\bar{z}} \cosh \alpha \end{array} Energy-momentum 4-vector  Charge density-current 4-vector E=E¯coshα+pz¯sinhαρ=ρ¯coshα+jz¯sinhαpx=px¯jx=jx¯py=py¯jy=jy¯pz=E¯sinhα+pz¯coshαjz=ρ¯sinhα+jz¯coshα
Aberration, incoming photon:
sin θ = p E = ( 1 β 2 ) 1 / 2 sin θ ¯ 1 β cos θ ¯ sin θ ¯ = p ¯ E ¯ = ( 1 β 2 ) 1 / 2 sin θ 1 + β cos θ cos θ = p z E = cos θ ¯ β 1 β cos θ ¯ cos θ ¯ = p ¯ z ¯ E ¯ = cos θ + β 1 + β cos θ tan ( θ / 2 ) = e α tan ( θ ¯ / 2 ) tan ( θ ¯ / 2 ) = e α tan ( θ / 2 ) sin θ = p E = 1 β 2 1 / 2 sin θ ¯ 1 β cos θ ¯ sin θ ¯ = p ¯ E ¯ = 1 β 2 1 / 2 sin θ 1 + β cos θ cos θ = p z E = cos θ ¯ β 1 β cos θ ¯ cos θ ¯ = p ¯ z ¯ E ¯ = cos θ + β 1 + β cos θ tan ( θ / 2 ) = e α tan ( θ ¯ / 2 ) tan ( θ ¯ / 2 ) = e α tan ( θ / 2 ) {:[sin theta=(-p_(_|_))/(E)=((1-beta^(2))^(1//2)sin ( bar(theta)))/(1-beta cos ( bar(theta)))sin bar(theta)=(- bar(p)_(_|_))/(( bar(E)))=((1-beta^(2))^(1//2)sin theta)/(1+beta cos theta)],[cos theta=(-p^(z))/(E)=(cos ( bar(theta))-beta)/(1-beta cos ( bar(theta)))cos bar(theta)=(- bar(p)^( bar(z)))/(( bar(E)))=(cos theta+beta)/(1+beta cos theta)],[tan(theta//2)=e^(alpha)tan( bar(theta)//2)tan( bar(theta)//2)=e^(-alpha)tan(theta//2)]:}\begin{aligned} \sin \theta & =\frac{-p_{\perp}}{E}=\frac{\left(1-\beta^{2}\right)^{1 / 2} \sin \bar{\theta}}{1-\beta \cos \bar{\theta}} & \sin \bar{\theta}=\frac{-\bar{p}_{\perp}}{\bar{E}}=\frac{\left(1-\beta^{2}\right)^{1 / 2} \sin \theta}{1+\beta \cos \theta} \\ \cos \theta & =\frac{-p^{z}}{E}=\frac{\cos \bar{\theta}-\beta}{1-\beta \cos \bar{\theta}} & \cos \bar{\theta}=\frac{-\bar{p}^{\bar{z}}}{\bar{E}}=\frac{\cos \theta+\beta}{1+\beta \cos \theta} \\ \tan (\theta / 2) & =e^{\alpha} \tan (\bar{\theta} / 2) & \tan (\bar{\theta} / 2)=e^{-\alpha} \tan (\theta / 2) \end{aligned}sinθ=pE=(1β2)1/2sinθ¯1βcosθ¯sinθ¯=p¯E¯=(1β2)1/2sinθ1+βcosθcosθ=pzE=cosθ¯β1βcosθ¯cosθ¯=p¯z¯E¯=cosθ+β1+βcosθtan(θ/2)=eαtan(θ¯/2)tan(θ¯/2)=eαtan(θ/2)

Combination of Two Boosts in Same Direction

β = β 1 + β 2 1 + β 1 β 2 or α = α 1 + α 2 . β = β 1 + β 2 1 + β 1 β 2  or  α = α 1 + α 2 . beta=(beta_(1)+beta_(2))/(1+beta_(1)beta_(2))quad" or "quad alpha=alpha_(1)+alpha_(2).\beta=\frac{\beta_{1}+\beta_{2}}{1+\beta_{1} \beta_{2}} \quad \text { or } \quad \alpha=\alpha_{1}+\alpha_{2} .β=β1+β21+β1β2 or α=α1+α2.

General Combinations of Boosts and Rotations

Spinor formalism of Chapter 41

Poincaré Transformation

x μ = Λ α μ x α + a μ . x μ = Λ α μ x α + a μ . x^(mu)=Lambda_(alpha^('))^(mu)x^(alpha^('))+a^(mu).x^{\mu}=\Lambda_{\alpha^{\prime}}^{\mu} x^{\alpha^{\prime}}+a^{\mu} .xμ=Λαμxα+aμ.
Condition on the Lorentz part of this transformation:
d s 2 = η α β d x α d x β = d s 2 = η μ ν Λ α μ Λ β ν d x α d x β d s 2 = η α β d x α d x β = d s 2 = η μ ν Λ α μ Λ β ν d x α d x β ds^('2)=eta_(alpha^(')beta^('))dx^(alpha^('))dx^(beta^('))=ds^(2)=eta_(mu nu)Lambda_(alpha^('))^(mu)Lambda_(beta^('))^(nu)dx^(alpha^('))dx^(beta^('))d s^{\prime 2}=\eta_{\alpha^{\prime} \beta^{\prime}} d x^{\alpha^{\prime}} d x^{\beta^{\prime}}=d s^{2}=\eta_{\mu \nu} \Lambda_{\alpha^{\prime}}^{\mu} \Lambda_{\beta^{\prime}}^{\nu} d x^{\alpha^{\prime}} d x^{\beta^{\prime}}ds2=ηαβdxαdxβ=ds2=ημνΛαμΛβνdxαdxβ
or Λ T η Λ = η Λ T η Λ = η Lambda^(T)eta Lambda=eta\Lambda^{T} \eta \Lambda=\etaΛTηΛ=η (matrix equation, with T T TTT indicating "transposed," or rows and columns interchanged).
Effect of transformation on other quantities:
u μ = Λ μ α u α α p μ = Λ μ α p α F μ ν = Λ μ α Λ ν β F α β e α = e μ Λ μ α ω α = Λ α μ ω μ u = e α u α = e μ u μ = u (4-velocity) (4-momentum) (electromagnetic field) u α = u μ Λ μ α ; p α = p μ Λ μ α ; F α β = F μ ν Λ α μ Λ β ; (basis vectors); (basis 1-forms); (the 4 -velocity vector). u μ = Λ μ α u α α p μ = Λ μ α p α F μ ν = Λ μ α Λ ν β F α β e α = e μ Λ μ α ω α = Λ α μ ω μ u = e α u α = e μ u μ = u  (4-velocity)   (4-momentum)   (electromagnetic field)  u α = u μ Λ μ α ; p α = p μ Λ μ α ; F α β = F μ ν Λ α μ Λ β ;  (basis vectors);   (basis 1-forms);   (the  4 -velocity vector).  {:[u^(mu)=Lambda^(mu)_(alpha)u^(alpha^(alpha))],[p^(mu)=Lambda^(mu)_(alpha)p^(alpha^('))],[F^(mu nu)=Lambda^(mu)_(alpha)Lambda^(nu)_(beta^('))F^(alpha^(')beta^('))],[e_(alpha^('))=e_(mu)Lambda^(mu)_(alpha^('))],[omega^(alpha^('))=Lambda^(alpha^('))_(mu)omega^(mu)],[u=e_(alpha^('))u^(alpha^('))=e_(mu)u^(mu)=u],[" (4-velocity) "],[" (4-momentum) "],[" (electromagnetic field) "],[u_(alpha^('))=u_(mu)Lambda^(mu)_(alpha^('));],[p_(alpha^('))=p_(mu)Lambda^(mu)_(alpha^('));],[F_(alpha^(')beta^('))=F_(mu nu)Lambda_(alpha^('))^(mu)Lambda_(beta^('));],[" (basis vectors); "],[" (basis 1-forms); "],[" (the "4"-velocity vector). "]:}\begin{aligned} & u^{\mu}=\Lambda^{\mu}{ }_{\alpha} u^{\alpha^{\alpha}} \\ & p^{\mu}=\Lambda^{\mu}{ }_{\alpha} p^{\alpha^{\prime}} \\ & F^{\mu \nu}=\Lambda^{\mu}{ }_{\alpha} \Lambda^{\nu}{ }_{\beta^{\prime}} F^{\alpha^{\prime} \beta^{\prime}} \\ & \boldsymbol{e}_{\alpha^{\prime}}=\boldsymbol{e}_{\mu} \Lambda^{\mu}{ }_{\alpha^{\prime}} \\ & \boldsymbol{\omega}^{\alpha^{\prime}}=\Lambda^{\alpha^{\prime}}{ }_{\mu} \boldsymbol{\omega}^{\mu} \\ & \boldsymbol{u}=\boldsymbol{e}_{\alpha^{\prime}} u^{\alpha^{\prime}}=\boldsymbol{e}_{\mu} u^{\mu}=\boldsymbol{u} \\ & \text { (4-velocity) } \\ & \text { (4-momentum) } \\ & \text { (electromagnetic field) } \\ & u_{\alpha^{\prime}}=u_{\mu} \Lambda^{\mu}{ }_{\alpha^{\prime}} ; \\ & p_{\alpha^{\prime}}=p_{\mu} \Lambda^{\mu}{ }_{\alpha^{\prime}} ; \\ & F_{\alpha^{\prime} \beta^{\prime}}=F_{\mu \nu} \Lambda_{\alpha^{\prime}}^{\mu} \Lambda_{\beta^{\prime}} ; \\ & \text { (basis vectors); } \\ & \text { (basis 1-forms); } \\ & \text { (the } 4 \text {-velocity vector). } \end{aligned}uμ=Λμαuααpμ=ΛμαpαFμν=ΛμαΛνβFαβeα=eμΛμαωα=Λαμωμu=eαuα=eμuμ=u (4-velocity)  (4-momentum)  (electromagnetic field) uα=uμΛμα;pα=pμΛμα;Fαβ=FμνΛαμΛβ; (basis vectors);  (basis 1-forms);  (the 4-velocity vector). 

Exercise 2.7. BOOST IN AN ARBITRARY DIRECTION

An especially useful Lorentz transformation has the matrix components
Λ 0 0 = γ 1 1 β 2 , Λ 0 j = Λ j 0 = β γ n j , (2.44) Λ j k = Λ k j = ( γ 1 ) n j n k + δ j k , Λ μ ν = ( same as Λ v μ but with β replaced by β ), Λ 0 0 = γ 1 1 β 2 , Λ 0 j = Λ j 0 = β γ n j , (2.44) Λ j k = Λ k j = ( γ 1 ) n j n k + δ j k , Λ μ ν =  same as  Λ v μ  but with  β  replaced by  β  ),  {:[Lambda^(0^('))_(0)=gamma-=(1)/(sqrt(1-beta^(2)))","],[Lambda^(0^('))_(j)=Lambda^(j^('))_(0)=-beta gamman^(j)","],[(2.44)Lambda^(j^('))_(k)=Lambda^(k^('))_(j)=(gamma-1)n^(j)n^(k)+delta^(jk)","],[Lambda^(mu)_(nu^('))=(" same as "Lambda^(v^('))_(mu)" but with "beta" replaced by "-beta:}" ), "]:}\begin{align*} & \Lambda^{0^{\prime}}{ }_{0}=\gamma \equiv \frac{1}{\sqrt{1-\beta^{2}}}, \\ & \Lambda^{0^{\prime}}{ }_{j}=\Lambda^{j^{\prime}}{ }_{0}=-\beta \gamma n^{j}, \\ & \Lambda^{j^{\prime}}{ }_{k}=\Lambda^{k^{\prime}}{ }_{j}=(\gamma-1) n^{j} n^{k}+\delta^{j k}, \tag{2.44}\\ & \Lambda^{\mu}{ }_{\nu^{\prime}}=\left(\text { same as } \Lambda^{v^{\prime}}{ }_{\mu} \text { but with } \beta \text { replaced by }-\beta\right. \text { ), } \end{align*}Λ00=γ11β2,Λ0j=Λj0=βγnj,(2.44)Λjk=Λkj=(γ1)njnk+δjk,Λμν=( same as Λvμ but with β replaced by β ), 
where β , n 1 , n 2 β , n 1 , n 2 beta,n^(1),n^(2)\beta, n^{1}, n^{2}β,n1,n2, and n 3 n 3 n^(3)n^{3}n3 are parameters, and n 2 ( n 1 ) 2 + ( n 2 ) 2 + ( n 3 ) 2 = 1 n 2 n 1 2 + n 2 2 + n 3 2 = 1 n^(2)-=(n^(1))^(2)+(n^(2))^(2)+(n^(3))^(2)=1\boldsymbol{n}^{2} \equiv\left(n^{1}\right)^{2}+\left(n^{2}\right)^{2}+\left(n^{3}\right)^{2}=1n2(n1)2+(n2)2+(n3)2=1. Show (a) that this does satisfy the condition Λ T η Λ = η Λ T η Λ = η Lambda^(T)eta Lambda=eta\Lambda^{T} \eta \Lambda=\etaΛTηΛ=η required of a Lorentz transformation (see Box 2.4); (b) that the primed frame moves with ordinary velocity β n β n beta n\beta \boldsymbol{n}βn as seen in the unprimed frame; (c) that the unprimed frame moves with ordinary velocity β n β n -beta n-\beta \boldsymbol{n}βn (i.e., v 1 = β n 1 , v 2 = β n 2 v 1 = β n 1 , v 2 = β n 2 v^(1^('))=-betan^(1),v^(2^('))=-betan^(2)v^{1^{\prime}}=-\beta n^{1}, v^{2^{\prime}}=-\beta n^{2}v1=βn1,v2=βn2, v 3 = β n 3 v 3 = β n 3 v^(3^('))=-betan^(3)v^{3^{\prime}}=-\beta n^{3}v3=βn3 ) as seen in the primed frame; and (d) that for motion in the z z zzz direction, the transformation matrices reduce to the familiar form
(2.45) A v μ = γ 0 0 β γ 0 1 0 0 0 0 1 0 β γ 0 0 γ , Λ p = γ 0 0 β γ 0 1 0 0 0 0 1 0 β γ 0 0 γ (2.45) A v μ = γ 0 0 β γ 0 1 0 0 0 0 1 0 β γ 0 0 γ , Λ p = γ 0 0 β γ 0 1 0 0 0 0 1 0 β γ 0 0 γ {:(2.45)||A^(v^('))_(mu)||=||[gamma,0,0,-beta gamma],[0,1,0,0],[0,0,1,0],[-beta gamma,0,0,gamma]||","quad||Lambda_(p^('))^(')||=||[gamma,0,0,beta gamma],[0,1,0,0],[0,0,1,0],[beta gamma,0,0,gamma]||:}\left\|A^{v^{\prime}}{ }_{\mu}\right\|=\left\|\begin{array}{rrrr} \gamma & 0 & 0 & -\beta \gamma \tag{2.45}\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ -\beta \gamma & 0 & 0 & \gamma \end{array}\right\|, \quad\left\|\Lambda_{p^{\prime}}{ }^{\prime}\right\|=\left\|\begin{array}{cccc} \gamma & 0 & 0 & \beta \gamma \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \beta \gamma & 0 & 0 & \gamma \end{array}\right\|(2.45)Avμ=γ00βγ01000010βγ00γ,Λp=γ00βγ01000010βγ00γ

§2.10. COLLISIONS

Whatever the physical entity, whether it is an individual mass in motion, or a torrent of fluid, or a field of force, or the geometry of space itself, it is described in classical general relativity as a geometric object of its own characteristic kind. Each such object is built directly or by abstraction from identifiable points, and needs no coordinates for its representation. It has been seen how this coordinate-free description translates into, and how it can be translated out of, the language of coordinates and components, and how components in a local Lorentz frame transform under a Lorentz transformation. Turn now to two elementary applications of this mathematical machinery to a mass in motion. One has to do with short-range forces (collisions, this section); the other, with the long-range electromagnetic force (Lorentz force law, next chapter).
In a collision, all the change in momentum is concentrated in a time that is short compared to the time of observation. Moreover, the target is typically so small, and quantum mechanics so dominating, that a probabilistic description is the right one. A quantity
(2.46) d σ = ( d σ d Ω ) θ d Ω (2.46) d σ = d σ d Ω θ d Ω {:(2.46)d sigma=((d sigma)/(d Omega))_(theta)d Omega:}\begin{equation*} d \sigma=\left(\frac{d \sigma}{d \Omega}\right)_{\theta} d \Omega \tag{2.46} \end{equation*}(2.46)dσ=(dσdΩ)θdΩ
gives the cross section ( cm 2 ) cm 2 (cm^(2))\left(\mathrm{cm}^{2}\right)(cm2) for scattering into the element of solid angle d Ω d Ω d Omegad \OmegadΩ at the deflection angle θ θ theta\thetaθ; a more complicated expression gives the probability that the

EXERCISE

Conservation of energy-momentum in a collision
original particle will enter the aperture d Ω d Ω d Omegad \OmegadΩ at a given polar angle θ θ theta\thetaθ and azimuth ϕ ϕ phi\phiϕ and with energy E E EEE to E + d E E + d E E+dEE+d EE+dE, while simultaneously products of reaction also emerge into specified energy intervals and into specified angular apertures. It would be out of place here to enter into the calculation of such cross sections, though it is a fascinating branch of atomic physics. It is enough to note that the cross section is an area oriented perpendicular to the line of travel of the incident particle. Therefore it is unaffected by any boost of the observer in that direction, provided of course that energies and angles of emergence of the particles are transformed in accordance with the magnitude of that boost ("same events seen in an altered reference system").
Over and above any such detailed account of the encounter as follows from the local dynamic analysis, there stands the law of conservation of energy-momentum:
(2.47) original particles, J p J = final particles, , K p K . (2.47)  original   particles,  J p J =  final   particles,  , K p K . {:(2.47)sum_({:[" original "],[" particles, "J]:})p_(J)=sum_({:[" final "],[" particles, "","K]:})p_(K).:}\begin{equation*} \sum_{\substack{\text { original } \\ \text { particles, } J}} \boldsymbol{p}_{J}=\sum_{\substack{\text { final } \\ \text { particles, }, K}} \boldsymbol{p}_{K} . \tag{2.47} \end{equation*}(2.47) original  particles, JpJ= final  particles, ,KpK.
Out of this relation, one wins without further analysis such simple results as the following. (1) A photon traveling as a plane wave through empty space cannot split (not true for a focused photon!). (2) When a high-energy electron strikes an electron at rest in an elastic encounter, and the two happen to come off sharing the energy equally, then the angle between their directions of travel is less than the Newtonian value of 90 90 90^(@)90^{\circ}90, and the deficit gives a simple measure of the energy of the primary. (3) When an electron makes a head-on elastic encounter with a proton, the formula for the fraction of kinetic energy transferred has three rather different limiting forms, according to whether the energy of the electron is nonrelativistic, relativistic, or extreme-relativistic. (4) The threshold for the production of an ( e + , e ) e + , e (e^(+),e^(-))\left(e^{+}, e^{-}\right)(e+,e)pair by a photon in the field of force of a massive nucleus is 2 m e 2 m e 2m_(e)2 m_{e}2me. (5) The threshold for the production of an ( e + , e ) e + , e (e^(+),e^(-))\left(e^{+}, e^{-}\right)(e+,e)pair by a photon in an encounter with an electron at rest is 4 m e 4 m e 4m_(e)4 m_{e}4me (or 4 m e ϵ 4 m e ϵ 4m_(e)-epsilon4 m_{e}-\epsilon4meϵ when account is taken of the binding of the e + e e e + e e e^(+)e^(-)e^(-)e^{+} e^{-} e^{-}e+eesystem in a very light "molecule"). All these results (topics for independent projects!) and more can be read out of the law of conservation of energy-momentum. For more on this topic, see Blaton (1950), Hagedorn (1964), and Chapter 4 and the last part of Chapter 5 of Sard (1970).

  1. Program in Science, Technology, and Society, and Department of Physics, Massachusetts Institute of Technology. Cambridge, Mass. dikaiser@mit.edu.
    Portions of this essay are adapted from David Kaiser, "A Tale of Two Textbooks: Experiments in Genre," Isis 103 (March 2012): 126-38.
  2. 1 1 ^(1){ }^{1}1 The following abbreviations are used in the notes: JAW, John A. Wheeler papers, American Philosophical Society, Philadelphia, Pennsylvania; KST, Kip S. Thorne papers, in Professor Thorne's possession, California Institute of Technology, Pasadena, California.
    2 2 ^(2){ }^{2}2 Charles W. Misner, Kip S. Thorne, and John A. Wheeler, Gravitation (San Francisco: W. H. Freeman, 1973). On nicknames for the book, see, e.g., "Chicago Undergraduate Physics Bibliography," available at http://www .ocf.berkeley.edu/~abhishek/chicphys.htm.
  3. 3 3 ^(3){ }^{3}3 For succinct introductions to the early history of Einstein's work on general relativity, see Michel Janssen, "'No success like failure': Einstein's quest for general relativity," in The Cambridge Companion to Einstein, ed. Michel Janssen and Christoph Lehner (New York: Cambridge University Press, 2014), 167-227; Hanoch Gutfreund and Jürgen Renn, The Road to Relativity: The History and Meaning of Einstein's "The Foundation of General Relativity" (Princeton, NJ: Princeton University Press, 2015); and Michel Janssen and Jürgen Renn, "Arch and scaffold: How Einstein found his field equations," Physics Today 68 (November 2015): 30-36.
    4 4 ^(4){ }^{4}4 Albert Einstein, "Foreword," in Peter G. Bergmann, Introduction to the Theory of Relativity (New York: Prentice-Hall, 1942), v. On Eddington's eclipse expedition and the early reception of general relativity, see Jean Eisenstaedt, The Curious History of Relativity: How Einstein's Theory of Gravity was Lost and Found Again (Princeton, NJ: Princeton University Press, 2006); Jeffrey Crelinstein, Einstein's Jury: The Race to Test Relativity (Princeton, NJ: Princeton University Press, 2006); Matthew Stanley, Practical Mystic: Religion, Science, and A. S. Eddington (Chicago: University of Chicago Press, 2007), chapter 3; and Hanoch Gutfreund and Jürgen Renn, The Formative Years of Relativity: The History and Meaning of Einstein's Princeton Lectures (Princeton, NJ: Princeton University Press, 2017).
    5 5 ^(5){ }^{5}5 On the return of general relativity to physics departments' course offerings during the 1950s and 1960s, see David Kaiser, "A psi is just a psi? Pedagogy, practice, and the reconstitution of general relativity, 1942-1975," Studies in the History and Philosophy of Modern Physics 29 (1998): 321-338; Daniel Kennefick, Traveling at the Speed of Thought: Einstein and the Quest for Gravitational Waves (Princeton, NJ: Princeton University Press, 2007), chapter 6; and Alexander Blum, Roberto Lalli, and Jürgen Renn, "The reinvention of general relativity: A historiographical framework for assessing one hundred years of curved space-time," Isis 106, no. 3 (September 2015): 598-620. On Wheeler as an effective mentor, see Charles W. Misner, Kip S. Thorne, and Wojciech H. Zurek, "John Wheeler, relativity, and quantum information," Physics Today 62 (April 2009): 40-46; and Terry M. Christensen, "John Wheeler's mentorship: An enduring legacy," Physics Today 62 (April 2009): 55-59.
    6 6 ^(6){ }^{6}6 Steven Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (New York: Wiley, 1972); S. W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time (New York: Cambridge University Press, 1973).
  4. 7 7 ^(7){ }^{7}7 John Wheeler, handwritten notes, "Thoughts on preface, Mon., 13 July 1970," in JAW Series IV, Box F-L, Folder, "Gravitation: Notes with Charles W. Misner and Kip S. Thorne" ("committee planning graduate courses"). See also form letter from Misner, Thorne, and Wheeler to colleagues announcing forthcoming publication of the book, 13 June 1973, in KST Folder "MTW: Sample pages."
    8 8 ^(8){ }^{8}8 John Wheeler, handwritten notes, page for insertion into draft of preface, n.d., late August 1970 ("third channel of pedagogy"); Wheeler, handwritten notes, "Plan of Book, Sat., 18 July 1970" ("test a write up"), both in JAW Series IV, Box F-L, Folder, "Gravitation: Notes with Charles W. Misner and Kip S. Thorne." (Emphasis in original.) On sidebars in more elementary physics textbooks, see Sharon Traweek, Beamtimes and Lifetimes: The World of High-Energy Physicists (Cambridge, MA: Harvard University Press, 1988), 76-81.
    9 9 ^(9){ }^{9}9 Kip Thorne to Earl Tondreau (editor at W. H. Freeman), 14 October 1970, in KST Folder "MTW: Correspondence, 1970-May, 1973" ("Several features," typefaces). See also Thorne to Robert Ishikawa and Aidan Kelley (W. H. Freeman), 28 January 1971, in KST Folder "MTW"; and Evan Gillespie (W. H. Freeman) to Kip Thorne, 29 November 1972, in KST Folder "MTW: Publishing company, 1970-71, 1971-72."
  5. 10 10 ^(10){ }^{10}10 Kip Thorne to Ya. B. Zel'dovich and I. D. Novikov, 21 June 1973, in KST Folder "MTW: Correspondence, June, 1973-."
    11 11 ^(11){ }^{11}11 Thorne to Ishikawa and Kelley, 28 January 1971 ("dependency statements").
    12 12 ^(12){ }^{12}12 Kip Thorne to John Wheeler and Charles Misner, with cc to Bruce Armbruster, 17 February 1972, in KST Folder, "MTW: Correspondence, 1970-May, 1973."
    13 13 ^(13){ }^{13}13 Thorne to Wheeler and Misner with cc to Armbruster, 17 February 1972. See also Misner, Thorne, and Wheeler, form letter to colleagues, 13 June 1973, in KST Folder, "MTW: Sample pages."
  6. 14 14 ^(14){ }^{14}14 Thorne to Bruce Armbruster, 10 April 1973 (royalty rates, pricing vis-à-vis Weinberg's book, "capture one hundred percent"), in KST Folder "MTW: Publishing company, 1970-71, 1971-72." On pricing, see also Thorne to Richard Warrington (president), Peter Renz (science editor), and Lew Kimmick (financial manager) at W. H. Freeman, 14 February 1979, in JAW Series II, Box Fr-Gl, Folder "W. H. Freeman and Co., Publishers"; Thorne to Wheeler and Misner, 2 November 1972, in KST Folder "MTW"; Misner to Wheeler and Thorne, 18 November 1982, in KST Folder "MTW" (copy also in JAW Series II, Box Fr-Gl, Folder "W. H. Freeman and Co., Publishers"); and royalty statement from June 1993 in KST Folder "MTW: Royalty statements."
    15 15 ^(15){ }^{15}15 Dennis Sciama, "Modern view of general relativity," Science 183 (March 22, 1974): 1186 ("pedagogic masterpiece"); Michael Berry, review in Science Progress 62, no. 246 (1975): 356-360, on 360 ("Aladdin's cave"); David Park, "Ups and downs of 'Gravitation,"" Washington Post (April 21, 1974): 4 ("three highly inventive people"). See also D. Allan Bromley, review in American Scientist (January-February 1974): 101-102.
    16 16 ^(16){ }^{16}16 L. Resnick, review in Physics in Canada (June 1975), clipping in KST Folder "MTW: Reviews" ("difficult book to read"); S. Chandrasekhar, "A vast treatise on general relativity," Physics Today (August 1974): 47-48, on 48 ("needless repetition"); W. H. McCrea, review in Contemporary Physics 15, no. 4 (July 1974), clipping in KST Folder "MTW: Reviews" ("variety of gimmicks").
    17 17 ^(17){ }^{17}17 John Wheeler, handwritten "Thoughts on preface, Mon. 13 July 1970," in JAW Series IV, Box F-L, Folder, "Gravitation: Notes with Charles W. Misner and Kip S. Thorne" ("make clear the idea"). On Wheeler's style, see also John A. Wheeler with Kenneth Ford, Geons, Black Holes, and Quantum Foam: A Life in Physics (New York: W. W. Norton, 1998); and Misner, Thorne, and Zurek, "John Wheeler, relativity, and quantum information."
  7. 18 18 ^(18){ }^{18}18 Sciama, "Modern view of general relativity," 1186 ("prose style"); Resnick, review in Physics in Canada ("commendable attempt"); J. Bicak, review in Bulletin of the Astronomical Institute of Czechoslovakia 26, no. 6 (1975): 377-378 ("A 'poetical' style").
    19 19 ^(19){ }^{19}19 Alan Farmer, review in Journal of the British Interplanetary Society 27 (1974): 314-315, on 314 ("comes dangerously close"); Ian Roxburgh, "Geometry is all, or is it?" New Scientist (September 26, 1974): 828 ("a regular subscriber").
    20 20 ^(20){ }^{20}20 Chandrasekhar, "A vast treatise on general relativity," 48; Thorne to Chandrasekhar, 21 June 1974, in KST Folder "MTW: Reviews." On Chandrasekhar's career, see K. C. Wali, Chandra: A Biography of S. Chandrasekhar (Chicago: University of Chicago Press, 1991); and Arthur I. Miller, Empire of the Stars: Obsession, Friendship, and Betrayal in the Quest for Black Holes (Boston: Houghton Mifflin, 2005).
    21 21 ^(21){ }^{21}21 Kip Thorne to Peter Renz, 15 June 1983, in KST Folder "MTW" ("large fraction of the physics graduate students"); Thorne to Warrington, Renz, and Kimmick, 14 February 1979, on annual sales of Gravitation and Weinberg's textbook.
    22 22 ^(22){ }^{22}22 Sales figures from royalty statement of June 1993 in KST Folder "MTW: Royalty statements." On PhD conferral rates, see David Kaiser, "Cold war requisitions, scientific manpower, and the production of American physicists after World War II," Historical Studies in the Physical and Biological Sciences 33 (2002): 131-159;
  8. and David Kaiser, "Booms, busts, and the world of ideas: Enrollment pressures and the challenge of specialization," Osiris 27 (2012): 276-302.
    23 23 ^(23){ }^{23}23 Kip Thorne to Peter Renz, 10 August 1983, in KST Folder "MTW."
    24 24 ^(24){ }^{24}24 Park, "Ups and downs of 'Gravitation," 4.
    25 25 ^(25){ }^{25}25 Robert Pincus, "Gravity theory excites the mind," clipping in KST Folder "MTW: Reviews." The clipping does not indicate date, publication title, or page number, but advertisements on the same page as the review clearly indicate that the newspaper was based in San Antonio, Texas.
    26 26 ^(26){ }^{26}26 See, e.g., Andrzej Trautman to Charles Misner, Kip Thorne, and John Wheeler, 10 January 1974, in KST Folder "MTW"; Heinz Pagels to Wheeler, 1 February 1974, in KST Folder "MTW Reviews"; Philip B. Burt to Wheeler, 12 November 1974, in KST Folder "MTW"; and Robert Rabinoff to Misner, Thorne, and Wheeler, 10 March 1978, in KST Folder "MTW: Reviews."
  9. 27 27 ^(27){ }^{27}27 Luigi Vignato to Charles Misner, Kip Thorne, and John Wheeler, 20 July 1976, in KST Folder "MTW: Correspondence, June, 1973-"; Wheeler to Vignato, 2 August 1976, in the same folder. Wheeler did not directly address Vignato's question, but he did enclose a preprint of his recent essay: John Wheeler, "Genesis and observership," in Foundational Problems in the Special Sciences, ed. Robert E. Butts and Jaakko Hintikka (Boston: Reidel, 1977), 3-33.
    28 28 ^(28){ }^{28}28 Jadoul Michel to Charles Misner, Kip Thorne, and John Wheeler, August 1983, in KST Folder "MTW."
    29 29 ^(29){ }^{29}29 Dan Foley to Kip Thorne, 7 February 1980, in KST Folder "MTW." See also Thorne to Foley, 27 February 1980, in the same folder.
  10. 30 30 ^(30){ }^{30}30 John Wheeler to Peter Renz, 28 June 1979, in KST Folder "MTW"; copy also in JAW Series II, Box Fr-Gl, Folder "W. H. Freeman and Co. Publishers."
  11. 1 1 ^(1){ }^{1}1 G. Holton (1965). 3 3 ^(3){ }^{3}3 A. Einstein (1949a).
    2 2 ^(2){ }^{2}2 L. Kolbros (1956). 4 4 ^(4){ }^{4}4 W. Thomson (1904).
    Citations for references will be found in the bibliography.
  12. 5 5 ^(5){ }^{5}5 As of April 1973, there are significant indications that Cygnus X-1 and other compact x-ray sources may be black holes.
  13. *This definition of a vector is valid only in flat spacetime. The refined definition ("tangent vector") in curved spacetime is not spelled out here (see Chapter 9), but flat-geometry ideas apply with good approximation even in a curved geometry, when the two points are sufficiently close.
    \dagger These formulas are precisely accurate only when the region of spacetime under consideration is flat and when in addition the coordinates are Lorentzian. Otherwise they are approximate-though they become arbitrarily good when the separation between points and the length of the vector become arbitrarily small.
  14. *"No difference" spelled out amounts to Einstein's (1911) principle of the local equivalence between a "gravitational field" and an acceleration: "We arrive at a very satisfactory interpretation of this law of experience, if we assume that the systems K and K K K^(')\mathrm{K}^{\prime}K are physically exactly equivalent, that is, if we assume that we may just as well regard the system K as being in a space free from gravitational fields, if we then regard K as uniformly accelerated. This assumption of exact physical equivalence makes it impossible for us to speak of the absolute acceleration of the system of reference, just as the usual theory of relativity forbids us to talk of the absolute velocity of a system; and it makes the equal falling of all bodies in a gravitational field seem a matter of course."
  15. Uffizi Gallery. Florence
  16. "He is not eternity or infinity, but eternal and infinite; He is not duration or space, but He endures and is present. He endures forever, and is everywhere present; and by existing always and everywhere, He constitutes duration and space. . . And thus much concerning God; to discourse of whom from the appearances of things, does certainly belong to natural philosophy."
    [FROM THE GENERAL SCHOLIUM AT THE END OF THE PRINCIPIA (1687)]
  17. *For example, see Goldstein (1959), Leighton (1959), Jackson (1962), or, for the physical perspective presented geometrically, Taylor and Wheeler (1966).